```python
from IPython.display import display, Math, Latex
import pystan
import matplotlib.pyplot as plt
import numpy as np
```

<http://www.DataJavelin.com>
## Dr Peter Hurley & Dr Phil Rooney

## Data Science

$$Data + Model = Prediction$$
* *Extracting information from Data*
* *Using information to make predictions, decisions etc*

 What is Probability?

 Idea: Probability is the Long Run Expected Number of Occurances of an Event

 When you roll a fair dice once it will come up with only one answer, but after doing this many times each outcome will occur 1/6 of the time. So the probability of an outcome is 1/6.

 Another way of saying this: The probability of an event A is equal to the number of ways in which A can occur divided by the number of ways in which any event can happen.

 This then helps us answer questions such as, what is the probability of drawing three aces in a row from a shuffled deck of cards?

 There are four ways of picking the first ace, out of 52 possible outcomes, three ways of picking the first ace, out of 51 possible outcomes, and two ways of picking the first ace, out of 50 possible outcomes. Therefore:

\begin{equation}
P(3\ Aces) = \frac{4}{52} * \frac{3}{51} * \frac{2}{50}
\end{equation}

 We can also ask more nuanced questions, such as what is the probability of event A given event B. For example, what is the probability of a card is a King given it is a face card?

\begin{equation}
P(A\ |\ B) =  \frac{P(A \cap	B)}{P(B)}
\end{equation}

\begin{equation}
P(King\ |\ Face\ Card) =  \frac{4}{12} =  \frac{1}{3}
\end{equation}

 Suppose a family has two children and suppose one of the children is a boy. What is the probability that both children are boys?

 Total number of ways in which a family can have two children:

 {BB}, {BG}, {GB}, {GG}

 Of these three involve at least one boy, and of these only one pairing is two boys so:

\begin{equation}
P(Two\ Boys\ |\ One\ child\ definitely\ a\ boy) =  \frac{1}{3}
\end{equation}

 A nice extension to this problem is: Suppose a family has two children and suppose one of the children is a boy born on a Tuesday. What is the probability that both children are boys?

 The Monte Hall Problem:Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

\begin{equation}
P(Prize\ behind\ any\ random\ door) =  \frac{1}{3}
\end{equation}

\begin{equation}
P(Keep\ Door\ 1\ and\ Win) =  \frac{P(Prize\ Door\ 1\ and\ Host\ Opens\ Door\ 3)}{Host\ Opens\ Door\ 3}
\end{equation}

\begin{equation}
P(Prize\ Door\ 1\ and\ Host\ Opens\ Door\ 3) =  \frac{1}{3} * \frac{1}{2}  =  \frac{1}{6}
\end{equation}

 The probability the host opens door 3 depends on which door the prize is behind.

\begin{equation}
P(Host\ Door\ 3) =  P(Host\ Door\ 3\ |\ Prize\ Door\ 1) * P(Prize\ Door\ 1) + P(Host\ Door\ 3\ |\ Prize\ Door\ 2) * P(Prize\ Door\ 3)+ P(Host\ Door\ 3\ |\ Prize\ Door\ 2) * P(Prize\ Door\ 3)
\end{equation}

\begin{equation}
P(Host\ Door\ 3) = \frac{1}{2} * \frac{1}{3} + 1 * \frac{1}{3} + 0*\frac{1}{3} = \frac{1}{2}
\end{equation}

 So the probability of winning if we keep our door is:

\begin{equation}
P(Keep\ Door\ 1\ and\ Win) = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}
\end{equation}

 And so the probability of winning if we swap is: <\p>

\begin{equation}
P(Swap\ Door\ 2\ and\ Win) = 1 - \frac{1}{3} =  \frac{2}{3}
\end{equation}

 We can manipulate these equations as in other areas of maths as we did above

\begin{equation}
P(A\ |\ B) =  \frac{P(A \cap	B)}{P(B)}
\end{equation}

\begin{equation}
P(A \cap	B) = P(A) * P(B | A)
\end{equation}

\begin{equation}
P(A \cap	B) = P(B) * P(A | B)
\end{equation}

\begin{equation}
P(A) * P(B | A) = P(B) * P(A | B)
\end{equation}

\begin{equation}
P(B | A) = \frac{P(B) * P(A | B)}{P(A)}
\end{equation}

 This last equation is Bayes Theorem, a source of heated debate among statisticians throughout the 20th century.

 You may wonder what the fuss is as surely everyone accepts the above?

 The issue comes in when we consider how far we can apply this equation.

## Probabilistic models
* A model describes data that one could observe from a system
* Uses probability theory to express all forms of uncertainty and noise associated with our model
* Inverse probability (i.e. Bayes rule) to infer parameters, adapt models, make predictions and learn from data.

A good example of how Bayesian probability is more intuitive is to think of a situation such as the probability of rain, given the sky is cloudy, or that there is a sprinkler nearby. We use prior information to decide if it has rained, rather than just looking at the number of times it has rained in past.

Alternatively there can also be cases where a frequentist model breaks down. Suppose you are an astronomer looking at Saturn with an old telescope. Combining many measurements of the shape of the planet will not improve your result on resolving the shape. However given a model of how a planet with a ring or without you can test how well each model describes Saturn and infer the correct shape.

Other key cases are those in which you can never repeat the experiment, e.g. modelling how the universe grew since the big bang or the diversification of songbirds in the Andes.

\begin{equation}
P(B | A) = \frac{P(B) * P(A | B)}{P(A)}
\end{equation}

 P(B | A) = Posterior

    P(B) = Prior

    P(A | B) = Likelihood

    P(A) = Marginal Likelihood

 Transparent way of including model prior information

Gives full probability distribution

 As the area under a posterior must equal one the marginal likelihood can be thought of as a normalistion value, and so the equation is often written as:

\begin{equation}
P(B | A) \propto P(B) * P(A | B)
\end{equation}

 In that case your posterior is your prior state of belief being updated by the new data.

 More precisely however you P(A) is actually an integral which is often difficult to solve, and must be done numerically,  motivates the need for MCMC type sampling. MCMC-like algorithms will generate points proportionally to the posterior values in each region. A simplistic way of thinking about this is a sampler generates some initial starting parameter values, calculates how well a model with these parameters explains the data and then generates a new set of parameter values nearby. If these new values are an improvement we always accept the new point and move there, otherwise we have a small chance we move there. The exact way in which this works varies by method.

 So far so good, but there is a lot of debate and discomfort with how you choose an appropriate prior. However disagreements over what prior is used can at least be transparent, and the correct description of the current state of knowledge before new data is an important and necessary part of debate.

### Why bother with all this Bayesian probabilistic stuff?
* Prior information help us extract more information from data

### Prob. model of straight line
$\mathbf{y}=m\mathbf{x}+c + \epsilon$

likelihood = $p(\mathbf{y}|m,c,\mathbf{x}) = \mathcal{N}(m\mathbf{x}+c,\sigma^2)$

prior = $p(m,c) = \mathcal{N}(0,\Sigma_p)$

posterior (after a bit of maths)

$p(m,c|\mathbf{y},\mathbf{x}) \propto  \mathcal{N}(\sigma^{-2}(\sigma^{-2}\mathbf{x}\mathbf{x}^T + \Sigma_{p}^{-1})^{-1}\mathbf{x}\mathbf{y},$
$\sigma^{-2}\mathbf{x}\mathbf{x}^T + \Sigma_{p}^{-1})$

Simple example straight line, error only in y, simple gaussian prior

What happens if intercept is positive? gets complicated fast

Have prior information on intercept

$c \sim \mathcal{N}(0.3,0.1)$

* The uncertainty in our inferred model parameters contains information.

* We do not want to just know the best answer, we
want to know how uncertain we are.

### Using Uncertianty in decision making means making better decisions

* Business consider risk when making decisions
* Uncertianty should be considered when making decisions based on predictions

See [Blog post on Bayesian Decision making](https://pdh21.github.io/jekyll/update/2019/06/06/FACYNation_BDT.html)

# Stan

* Language for building probabilistic models
* Algorithms for fitting probabilistic models
* Interfaces for using probabilistic models

## Language
Specify probabilisitic model: the joint probability distribution function,

$$p(\theta,x)$$

* $\theta$ = parameters
* $x$ = data
* $p$ = model

Stan is a language:
* Statically typed, imperative
* User can define programs using parameters, data and  $\log p(\theta,x)$
* Any program can be written as long as $\log p(\theta,x)$ is differentiable

Statically typed like C, C++. i.e. types declared at compile time rather than run time like Python.

Imperative, you tell the compiler what you want to happen, step by step

## Algorithms
* Bayesian Inference: No U-Turn Hamiltonian Monte Carlo,
    - $p(\theta|x)$ approximated with $[\theta^1,\theta^2,..\theta^N]$
* Approximate Bayesian Inference: Variational Inference,
    - $\hat{p}(\theta|x) \approx q(\hat{\phi})$, where $\phi = argmin_{\phi} D_{KL}(q(\theta|\phi)||p(\theta,x))$
* Optimisation: Max
    - $\hat{\theta} = argmax_{\theta} p(\theta,x)$

## Interfaces
* CmdStan,PyStan,RStan
* C++ API
* C++ auto-diff library
* Software built with Stan: RStanArm, brms, prophet

 As an example lets consider a question of if a coin is fair or not. We begin with a flat prior and update it as we flip a coin.

```python
coin_model ="""// Inferring a Rate
data {
  int<lower=1> n;
  int<lower=0> h;
}
parameters {
  real<lower=0,upper=1> theta;
}
model {
  // Prior Distribution for Rate Theta
  theta ~ normal(3, 0.5);

  // Observed Counts
  h ~ binomial(n, theta);
}"""

sm=pystan.StanModel(model_code=coin_model)
```

    INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_f5e1c19b675a739c99f9dc1c95646755 NOW.

```python
model_data={
        'n':100,
        'h':55,
}

```

```python
fit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)
```

```python
print(fit)

plt.plot(np.arange(0, fit['theta'].size), fit['theta'])
plt.xlabel('Sample')
plt.ylabel('Theta')
plt.show()

plt.hist(fit['theta'])
plt.xlabel('Theta')
plt.ylabel('Number of Samples')
plt.show
```

```python
model_data={
        'n':10,
        'h':7,
}

fit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)

print(fit)

plt.hist(fit['theta'])
plt.xlabel('Theta')
plt.ylabel('Number of Samples')
plt.show
```

```python
model_data={
        'n':100,
        'h':54,
}

fit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)

print(fit)

plt.hist(fit['theta'])
plt.xlabel('Theta')
plt.ylabel('Number of Samples')
plt.show
```

 Sometimes our data may come in a more complex form. For instance imagine if you have a game that involves coin tosses. You filp a coin until you flip a heads and a tails in a row. Your opponent flips a coin until they flip two heads in a row. All the information you have is who won each game. From this you want to understand if one coin was biased or not.

 As HT and HH in two coin flips are equally likely we may run the code above and see how close our theta value is to 0.5

 After 2,000 flips your opponent has won 747 times but you have won 1,253.

```python
model_data={
        'n':2000,
        'h':747,
}

fit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)

print(fit)

plt.hist(fit['theta'])
plt.xlabel('Theta')
plt.ylabel('Number of Samples')
plt.show
```

 So can we conclusively say that our coin was biased?

 Once again this is a case where we should have questioned our underlying assumptions, i.e. that both opponents had a 50% chance of winning. In future this will be covered in part by prior predictive checks which should give us intuition on how our model behaves. For now let us first generate data in the way described assuming both players have a fair coin

```python
bob_wins = 0
alice_wins = 0

for i in range(2000):
    bob_seq = np.array([])
    alice_seq = np.array([])
    bob_poss_win = 0
    alice_poss_win = 0
    while bob_wins + alice_wins < 2000:
        bob_seq = np.append(bob_seq, np.random.randint(2,size=1))
        alice_seq = np.append(alice_seq, np.random.randint(2,size=1))
        if np.prod((bob_seq.size > 1) and (bob_seq[-2:] == [0,0])):
            bob_poss_win = 1
        if np.prod((alice_seq.size > 1) and (alice_seq[-2:] == [0,1])):
            alice_poss_win = 1
        if (bob_poss_win == 1) and (alice_poss_win == 1):
            bob_seq = np.array([])
            alice_seq = np.array([])
            bob_poss_win = 0
            alice_poss_win = 0
        elif bob_poss_win == 1:
            bob_wins += 1
            bob_seq = np.array([])
            alice_seq = np.array([])
            bob_poss_win = 0
        elif alice_poss_win == 1:
            alice_wins += 1
            bob_seq = np.array([])
            alice_seq = np.array([])
            alice_poss_win = 0

print(bob_wins)
print(alice_wins)

```

    752
    1248

 Understanding a problem deeply is important when building a model. Misunderstanding a model can lead to misinterpretation. In this case there is a subtle difference, if you need to flip HT and first flip a H you will not win if on the next coin toss you flip another H - but you are still only one flip away. If you need to flip HH and after one H toss flip a T, you are again two flips away from winning.

## Probabilistic Programming
*Define probability models & solve automatically*

```python
rate_diff = """// Difference Between Two Rates
data {
  int<lower=1> n1;
  int<lower=1> n2;
  int<lower=0> k1;
  int<lower=0> k2;
}
parameters {
  real<lower=0,upper=1> theta1;
  real<lower=0,upper=1> theta2;
}
transformed parameters {
  real<lower=-1,upper=1> delta;
  delta <- theta1 - theta2;
}
model {
  // Prior Distribution for Rate Theta
  theta1 ~ beta(1, 1);
  theta2 ~ beta(1, 1);
  // Observed Counts
  k1 ~ binomial(n1, theta1);
  k2 ~ binomial(n2, theta2);
}"""

sm=pystan.StanModel(model_code=rate_diff)
```

    INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_b712d3d58d6ce64ce339b400e820a652 NOW.

```python
model_data = {
    'k1': 50,
    'k2': 62,
    'n1': 100,
    'n2': 100,
}

fit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)

plt.hist(fit['delta'])
plt.show()
```

 So let us now consider a model where there are more than one unknown and interacting parameters. In this case of five sets of eager volunteers distributed an equal number of surveys to randomly selected houses. Unfortunately nobody recorded the number of surveys in each bundle and now we need to estimate the rate of return. Assuming there was no more than 500 surveys in a bundle we can attempt to calculate this. But first let us discuss how we think the posterior will look.

```python
plt.plot(np.arange(0, 20), 1./np.arange(0, 20))
```

 # Inferring Return Rate and Numbers of Surveys from Observed Returns

# Observed Returns

for (i in 1:m){

k[i] ~ dbin(theta,n)

}

\# Priors on Rate Theta and Number n

theta ~ dbeta(1,1)

n ~ dcat(p[])

for (i in 1:nmax){

p[i] <- 1/nmax

}

```python
survey_model = """data {
  int<lower=0> nmax;
  int<lower=0> m;
  int<lower=0,upper=nmax> k[m];
}
transformed data {
  int<lower=0> nmin;  // Minimal possible n

  nmin <- max(k);
}
parameters {
  real<lower=0,upper=1> theta;
}
transformed parameters {
  vector[nmax] lp_parts;  // Log probability for each n

  // First part of the trick for mixture model
  for (n in 1:nmax)
    if (n < nmin)
      lp_parts[n] <- log(1.0 / nmax) + negative_infinity();  // Zero probability
    else
      lp_parts[n] <- log(1.0 / nmax) + binomial_log(k, n, theta);
}
model {
  // Second part of the trick for mixture model
  increment_log_prob(log_sum_exp(lp_parts));
}
generated quantities {
  int<lower=1,upper=nmax> n;
  simplex[nmax] prob_n;

  // Transforming lp_parts to probabilities of each n
  prob_n <- softmax(lp_parts);
  n <- categorical_rng(prob_n);
}"""

sm=pystan.StanModel(model_code=survey_model)
```

    INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_78f7cfc1daf3a7ccdfbee88500ff4157 NOW.

!!! DISCLAIMER !!!

Stan is not the correct tool to use if you are working with discrete parameters - in this case n. Expect errors and do this using a more suitable tool for real world applications.

```python
model_data={
    'k': np.array([16, 18, 22, 25, 27]),
    'm': 5,
    'nmax': 500
}

fit=sm.sampling(data=model_data,chains=4,iter=1000,seed=12329)

print(fit)

```

    WARNING:pystan:Maximum (flat) parameter count (1000) exceeded: skipping diagnostic tests for n_eff and Rhat.
    To run all diagnostics call pystan.check_hmc_diagnostics(fit)
    WARNING:pystan:2 of 2000 iterations ended with a divergence (0.1 %).
    WARNING:pystan:Try running with adapt_delta larger than 0.8 to remove the divergences.
    /opt/conda/lib/python3.7/site-packages/numpy/core/_methods.py:117: RuntimeWarning: invalid value encountered in subtract
      x = asanyarray(arr - arrmean)

    Inference for Stan model: anon_model_78f7cfc1daf3a7ccdfbee88500ff4157.
    4 chains, each with iter=1000; warmup=500; thin=1;
    post-warmup draws per chain=500, total post-warmup draws=2000.

                     mean se_mean      sd     2.5%      25%     50%      75%   97.5%  n_eff   Rhat
    theta            0.19  6.7e-3    0.14     0.05     0.08    0.15     0.26    0.55    429    1.0
    lp_parts[1]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[2]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[3]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[4]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[5]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[6]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[7]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[8]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[9]      -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[10]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[11]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[12]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[13]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[14]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[15]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[16]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[17]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[18]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[19]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[20]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[21]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[22]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[23]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[24]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[25]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[26]     -inf     nan     nan     -inf     -inf    -inf     -inf    -inf    nan    nan
    lp_parts[27]   -170.3    3.79   73.24   -291.3   -232.4  -168.5   -109.9   -44.1    373    1.0
    lp_parts[28]   -162.5    3.75   72.36   -282.5   -223.9  -160.3   -102.4  -39.05    372    1.0
    lp_parts[29]   -155.8    3.71   71.48   -274.9   -216.4  -153.3   -96.12  -35.15    372    1.0
    lp_parts[30]   -149.9    3.66    70.6   -268.1   -209.8  -147.0   -90.55  -31.99    371    1.0
    lp_parts[31]   -144.5    3.62   69.73   -261.8   -203.6  -141.2   -85.55  -29.41    371    1.0
    lp_parts[32]   -139.6    3.58   68.86   -255.9   -198.0  -136.0   -81.01  -27.62    370    1.0
    lp_parts[33]   -135.1    3.54    68.0   -250.5   -192.7  -131.1   -76.86  -26.13    370    1.0
    lp_parts[34]   -130.9    3.49   67.14   -245.4   -187.8  -126.6   -73.05  -25.08    369    1.0
    lp_parts[35]   -127.0    3.45   66.28   -240.6   -183.2  -122.3   -69.52  -23.99    369    1.0
    lp_parts[36]   -123.3    3.41   65.43   -236.0   -178.8  -118.3   -66.25   -23.6    368    1.0
    lp_parts[37]   -119.9    3.37   64.59   -231.7   -174.7  -114.5    -63.2  -22.69    368    1.0
    lp_parts[38]   -116.7    3.33   63.75   -227.6   -170.7  -111.0   -60.36  -22.16    367    1.0
    lp_parts[39]   -113.6    3.28   62.92   -223.6   -166.9  -107.6    -57.7  -21.81    367    1.0
    lp_parts[40]   -110.8    3.24   62.09   -219.8   -163.3  -104.4   -55.21  -21.62    366    1.0
    lp_parts[41]   -108.1     3.2   61.27   -216.2   -159.9  -101.3   -52.92  -21.56    366    1.0
    lp_parts[42]   -105.5    3.16   60.45   -212.7   -156.6  -98.42    -50.8  -21.37    365    1.0
    lp_parts[43]   -103.0    3.12   59.65   -209.4   -153.4  -95.63   -48.84   -21.3    365    1.0
    lp_parts[44]   -100.7    3.08   58.85   -206.2   -150.4  -92.96   -47.07  -21.19    365    1.0
    lp_parts[45]   -98.57    3.04   58.05   -203.0   -147.5  -90.41    -45.4  -21.05    364    1.0
    lp_parts[46]   -96.48     3.0   57.27   -200.0   -144.6  -87.97   -43.88  -20.95    364    1.0
    lp_parts[47]   -94.48    2.96   56.49   -197.1   -141.9  -85.62   -42.46  -20.91    363    1.0
    lp_parts[48]   -92.58    2.92   55.72   -194.3   -139.3   -83.4   -41.04  -20.88    363    1.0
    lp_parts[49]   -90.76    2.89   54.96   -191.6   -136.7  -81.29   -39.67  -20.84    363    1.0
    lp_parts[50]   -89.03    2.85    54.2   -188.9   -134.3  -79.32   -38.37  -20.77    363    1.0
    lp_parts[51]   -87.37    2.81   53.46   -186.4   -131.9  -77.31   -37.44  -20.71    362    1.0
    lp_parts[52]   -85.79    2.77   52.73   -183.9   -129.6  -75.45    -36.6  -20.65    362    1.0
    lp_parts[53]   -84.27    2.73    52.0   -181.4   -127.3  -73.66   -36.17  -20.62    362    1.0
    lp_parts[54]   -82.83     2.7   51.29   -179.1   -125.1  -71.94   -35.14  -20.57    362    1.0
    lp_parts[55]   -81.44    2.66   50.59   -176.8   -123.1  -70.35   -34.18  -20.59    361    1.0
    lp_parts[56]   -80.11    2.63   49.89   -174.6   -121.0  -68.75   -33.51  -20.56    361    1.0
    lp_parts[57]   -78.85    2.59   49.21   -172.4   -119.1  -67.18   -32.85  -20.55    361    1.0
    lp_parts[58]   -77.63    2.56   48.55   -170.2   -117.6  -65.94   -32.38  -20.55    361    1.0
    lp_parts[59]   -76.47    2.52   47.89   -168.2   -115.7  -64.78    -31.8  -20.54    361   1.01
    lp_parts[60]   -75.35    2.49   47.25   -166.1   -114.1  -63.41    -31.5  -20.52    361   1.01
    lp_parts[61]   -74.29    2.46   46.63   -164.2   -112.3  -61.97   -30.96  -20.52    361   1.01
    lp_parts[62]   -73.27    2.42   46.01   -162.5   -110.8  -60.84   -30.65  -20.53    361   1.01
    lp_parts[63]   -72.29    2.39   45.42   -160.6   -109.0  -59.86   -30.15  -20.55    361   1.01
    lp_parts[64]   -71.36    2.36   44.83   -158.7   -107.5   -59.2   -29.98  -20.57    361   1.01
    lp_parts[65]   -70.46    2.33   44.27   -157.2   -106.0   -58.1   -29.62  -20.55    361   1.01
    lp_parts[66]    -69.6     2.3   43.72   -155.7   -104.7  -57.73   -29.31  -20.53    361   1.01
    lp_parts[67]   -68.78    2.27   43.19   -153.9   -103.2  -56.63   -28.88  -20.54    362   1.01
    lp_parts[68]    -68.0    2.24   42.68   -152.3   -101.7  -55.75   -28.91  -20.52    362   1.01
    lp_parts[69]   -67.25    2.21   42.18   -150.6   -100.4  -54.86   -28.67  -20.49    363   1.01
    lp_parts[70]   -66.53    2.19   41.71   -149.1   -99.17  -54.42   -28.41  -20.51    363   1.01
    lp_parts[71]   -65.85    2.16   41.25   -147.5   -97.94  -53.46   -28.47   -20.5    364   1.01
    lp_parts[72]    -65.2    2.14   40.82   -146.0   -96.55  -53.29   -28.36  -20.49    365   1.01
    lp_parts[73]   -64.57    2.11   40.41   -144.5   -95.18  -52.82   -28.29  -20.49    366   1.01
    lp_parts[74]   -63.98    2.09   40.02   -143.3   -94.38  -52.27   -28.13  -20.49    367   1.01
    lp_parts[75]   -63.41    2.07   39.66   -143.0   -93.31   -51.7    -28.0  -20.49    368   1.01
    lp_parts[76]   -62.87    2.04   39.31   -141.5   -92.29  -50.98   -27.97  -20.49    370   1.01
    lp_parts[77]   -62.36    2.02   38.99   -140.1   -91.91  -50.76   -27.93  -20.49    371   1.01
    lp_parts[78]   -61.87     2.0    38.7   -139.2   -90.97  -49.86   -27.83  -20.47    373   1.01
    lp_parts[79]   -61.41    1.99   38.43   -138.1    -90.0  -49.65    -27.7  -20.46    374   1.01
    lp_parts[80]   -60.97    1.97   38.19   -136.9   -89.07  -49.07   -27.52  -20.46    376   1.01
    lp_parts[81]   -60.56    1.95   37.97   -135.7   -88.03   -49.1   -27.44  -20.46    378   1.01
    lp_parts[82]   -60.16    1.94   37.78   -134.8   -87.37  -49.07   -27.49  -20.46    380   1.01
    lp_parts[83]   -59.79    1.92   37.62   -133.7   -86.91  -49.03    -27.3  -20.48    382   1.01
    lp_parts[84]   -59.44    1.91   37.49   -132.4   -85.94  -48.53   -27.11  -20.46    385   1.01
    lp_parts[85]   -59.11     1.9   37.38   -131.1   -85.16  -48.08   -27.13  -20.46    387   1.01
    lp_parts[86]    -58.8    1.89    37.3   -130.3   -84.14  -47.72   -26.94  -20.46    390   1.01
    lp_parts[87]   -58.51    1.88   37.25   -129.3   -83.41  -47.13   -26.94  -20.46    393   1.01
    lp_parts[88]   -58.24    1.87   37.23   -129.7   -82.52  -46.89   -26.67  -20.45    395   1.01
    lp_parts[89]   -57.99    1.87   37.24   -128.6   -82.42  -47.01    -26.8  -20.46    398   1.01
    lp_parts[90]   -57.76    1.86   37.27   -128.7   -82.41  -46.67   -26.72  -20.47    401   1.01
    lp_parts[91]   -57.54    1.86   37.34   -129.9   -82.05  -46.67   -26.42  -20.48    404   1.01
    lp_parts[92]   -57.34    1.86   37.43   -130.5   -81.14   -46.3   -26.39  -20.46    407   1.01
    lp_parts[93]   -57.16    1.85   37.55   -130.9   -80.64  -46.11   -26.57  -20.45    410   1.01
    lp_parts[94]   -56.99    1.85    37.7   -133.8   -80.36  -46.03   -26.51  -20.46    413   1.01
    lp_parts[95]   -56.84    1.86   37.87   -136.7   -79.59  -45.79   -26.41  -20.47    416   1.01
    lp_parts[96]    -56.7    1.86   38.07   -138.3   -78.93  -45.78   -26.42  -20.47    419   1.01
    lp_parts[97]   -56.58    1.87    38.3   -137.2   -78.75  -45.77   -26.45  -20.45    422   1.01
    lp_parts[98]   -56.48    1.87   38.56   -136.1   -78.36   -45.5   -26.48  -20.47    425   1.01
    lp_parts[99]   -56.39    1.88   38.84   -135.1   -78.23  -45.28   -26.27  -20.47    427   1.01
    lp_parts[100]  -56.31    1.89   39.14   -134.9   -77.88  -45.14   -26.29  -20.47    430   1.01
    lp_parts[101]  -56.25     1.9   39.47   -136.6   -77.86  -45.19   -26.21  -20.45    433   1.01
    lp_parts[102]   -56.2    1.91   39.83   -139.3   -77.35  -45.09   -26.08  -20.45    436   1.01
    lp_parts[103]  -56.16    1.92    40.2   -142.1    -76.9   -45.3   -26.08  -20.45    438   1.01
    lp_parts[104]  -56.14    1.93    40.6   -144.8   -77.38   -45.2   -26.08  -20.45    441   1.01
    lp_parts[105]  -56.13    1.95   41.03   -147.6   -76.63  -44.94   -26.15  -20.45    443   1.01
    lp_parts[106]  -56.13    1.96   41.47   -150.4   -76.56  -44.48   -26.29  -20.45    446   1.01
    lp_parts[107]  -56.14    1.98   41.93   -153.2   -75.81  -44.24   -26.12  -20.45    448   1.01
    lp_parts[108]  -56.17     2.0   42.42   -156.0   -75.52  -44.26   -26.01  -20.45    450   1.01
    lp_parts[109]  -56.21    2.02   42.92   -158.9   -75.28   -44.0   -25.87  -20.45    451   1.01
    lp_parts[110]  -56.26    2.04   43.44   -161.7   -75.69  -43.74   -25.94  -20.44    453   1.01
    lp_parts[111]  -56.32    2.06   43.98   -164.6   -75.17  -43.55   -25.89  -20.44    454   1.01
    lp_parts[112]  -56.39    2.09   44.54   -167.4   -75.05  -43.67   -25.84  -20.44    456   1.01
    lp_parts[113]  -56.47    2.11   45.11   -170.3   -74.64  -43.73   -25.92  -20.44    457   1.01
    lp_parts[114]  -56.56    2.14    45.7   -173.2   -74.64  -43.56   -25.96  -20.44    458   1.01
    lp_parts[115]  -56.66    2.16    46.3   -176.1   -74.43  -43.22   -26.08  -20.45    459   1.01
    lp_parts[116]  -56.77    2.19   46.92   -179.0   -74.32   -43.1   -25.93  -20.45    460   1.01
    lp_parts[117]  -56.89    2.22   47.56   -181.9   -73.87   -43.1   -26.08  -20.45    460   1.01
    lp_parts[118]  -57.03    2.25    48.2   -184.8   -73.91  -43.14   -26.04  -20.45    461   1.01
    lp_parts[119]  -57.17    2.28   48.86   -187.8   -74.02  -43.16   -26.03  -20.45    461   1.01
    lp_parts[120]  -57.32    2.31   49.54   -190.7    -73.7  -43.07   -26.05  -20.46    462   1.01
    lp_parts[121]  -57.48    2.34   50.22   -193.6   -73.26  -43.33   -26.02  -20.45    462   1.01
    lp_parts[122]  -57.65    2.37   50.92   -196.6   -73.13  -43.19   -25.98  -20.45    463   1.01
    lp_parts[123]  -57.82     2.4   51.63   -199.6   -73.09   -43.3   -25.88  -20.45    463   1.01
    lp_parts[124]  -58.01    2.43   52.35   -202.6    -73.1  -43.22   -25.95  -20.46    463   1.01
    lp_parts[125]   -58.2    2.47   53.08   -205.5   -72.88  -43.05   -26.11  -20.46    463   1.01
    lp_parts[126]  -58.41     2.5   53.81   -208.5   -73.02   -42.9   -26.22  -20.46    464   1.01
    lp_parts[127]  -58.62    2.53   54.56   -211.5   -72.65  -42.83   -26.37  -20.46    464   1.01
    lp_parts[128]  -58.84    2.57   55.32   -214.6   -72.46  -42.77   -26.34  -20.46    464   1.01
    lp_parts[129]  -59.06     2.6   56.08   -217.6    -72.2  -42.64   -26.11  -20.47    464   1.01
    lp_parts[130]   -59.3    2.64   56.86   -220.6    -72.3  -42.42   -26.04  -20.48    464   1.01
    lp_parts[131]  -59.54    2.68   57.64   -223.6   -72.39  -42.22   -26.01  -20.47    464   1.01
    lp_parts[132]  -59.79    2.71   58.43   -226.7   -71.95  -42.07   -26.09  -20.45    464   1.01
    lp_parts[133]  -60.05    2.75   59.23   -229.7   -71.53  -42.17   -26.03  -20.46    464   1.01
    lp_parts[134]  -60.31    2.79   60.03   -232.8   -71.38   -42.5   -25.97  -20.46    464   1.01
    lp_parts[135]  -60.59    2.83   60.84   -235.9   -71.09  -42.76    -25.9  -20.46    464   1.01
    lp_parts[136]  -60.86    2.86   61.66   -238.9   -71.15   -42.9   -25.89  -20.45    464   1.01
    lp_parts[137]  -61.15     2.9   62.48   -242.0   -71.14  -42.63   -25.92  -20.45    464   1.01
    lp_parts[138]  -61.44    2.94   63.31   -245.1   -70.82  -42.63   -25.92  -20.44    463   1.01
    lp_parts[139]  -61.74    2.98   64.15   -248.2   -70.33  -42.66   -25.93  -20.45    463   1.01
    lp_parts[140]  -62.05    3.02   64.99   -251.3   -70.33  -42.58   -25.92  -20.46    463   1.01
    lp_parts[141]  -62.36    3.06   65.84   -254.4   -70.09  -42.61   -25.86  -20.46    463   1.01
    lp_parts[142]  -62.68     3.1   66.69   -257.5   -69.69  -42.65   -25.83  -20.46    463   1.01
    lp_parts[143]  -63.01    3.14   67.55   -260.6   -69.49  -42.48   -25.78  -20.45    463   1.01
    lp_parts[144]  -63.34    3.18   68.41   -263.7    -69.4  -42.32   -25.89  -20.45    462   1.01
    lp_parts[145]  -63.68    3.22   69.27   -266.9   -69.04  -42.46   -25.84  -20.46    462   1.01
    lp_parts[146]  -64.02    3.26   70.14   -270.0    -69.3   -42.6   -25.84  -20.45    462   1.01
    lp_parts[147]  -64.37     3.3   71.02   -273.2   -69.47  -42.73   -25.84  -20.45    462   1.01
    lp_parts[148]  -64.72    3.35    71.9   -276.3   -69.08   -42.7   -25.91  -20.45    462   1.01
    lp_parts[149]  -65.09    3.39   72.78   -279.5   -68.93  -42.48   -26.06  -20.45    461   1.01
    lp_parts[150]  -65.45    3.43   73.67   -282.6   -68.59   -42.3   -25.93  -20.45    461   1.01
    lp_parts[151]  -65.83    3.47   74.56   -285.8   -68.62  -42.45   -26.12  -20.46    461   1.01
    lp_parts[152]   -66.2    3.51   75.45   -288.9   -68.93  -42.51   -26.07  -20.47    461   1.01
    lp_parts[153]  -66.59    3.56   76.35   -292.1   -68.75  -42.72   -26.01  -20.47    461   1.01
    lp_parts[154]  -66.98     3.6   77.25   -295.3    -68.7  -42.71    -26.0  -20.47    460   1.01
    lp_parts[155]  -67.37    3.64   78.15   -298.5   -68.53   -42.8   -25.96  -20.48    460   1.01
    lp_parts[156]  -67.77    3.69   79.06   -301.7   -68.56   -42.6   -25.85  -20.49    460   1.01
    lp_parts[157]  -68.17    3.73   79.97   -304.9   -68.31  -42.49    -25.9  -20.48    460   1.01
    lp_parts[158]  -68.58    3.77   80.88   -308.1    -68.6  -42.43   -25.91  -20.48    459   1.01
    lp_parts[159]   -69.0    3.82    81.8   -311.3   -68.45  -42.59   -25.83  -20.47    459   1.01
    lp_parts[160]  -69.42    3.86   82.71   -314.5    -68.9  -42.65   -25.84  -20.48    459   1.01
    lp_parts[161]  -69.84    3.91   83.64   -317.7   -68.86  -42.76   -25.91  -20.48    459   1.01
    lp_parts[162]  -70.27    3.95   84.56   -320.9    -68.7  -42.86   -25.91  -20.48    458   1.01
    lp_parts[163]  -70.71    3.99   85.48   -324.2   -68.69  -43.01   -25.94  -20.47    458   1.01
    lp_parts[164]  -71.15    4.04   86.41   -327.4   -69.13  -43.01   -25.89  -20.48    458   1.01
    lp_parts[165]  -71.59    4.08   87.34   -330.6   -69.09  -43.03   -25.95  -20.47    458   1.01
    lp_parts[166]  -72.04    4.13   88.28   -333.9   -69.45  -43.09   -25.86  -20.47    458   1.01
    lp_parts[167]  -72.49    4.17   89.21   -337.1   -69.38  -42.77   -25.92  -20.48    457   1.01
    lp_parts[168]  -72.95    4.22   90.15   -340.4    -70.1  -42.94   -25.93  -20.49    457   1.01
    lp_parts[169]  -73.41    4.26   91.08   -343.6   -70.25  -42.92   -25.82   -20.5    457   1.01
    lp_parts[170]  -73.87    4.31   92.03   -346.9   -70.78   -42.9   -25.86   -20.5    457   1.01
    lp_parts[171]  -74.34    4.35   92.97   -350.1   -71.56  -43.03   -25.82  -20.49    456   1.01
    lp_parts[172]  -74.82     4.4   93.91   -353.4   -72.35  -42.96   -25.89  -20.49    456   1.01
    lp_parts[173]   -75.3    4.44   94.86   -356.7   -73.24  -42.83   -26.03  -20.48    456   1.01
    lp_parts[174]  -75.78    4.49   95.81   -360.0   -74.02  -42.73    -26.0  -20.49    456   1.01
    lp_parts[175]  -76.26    4.53   96.76   -363.2   -74.91  -42.66   -26.03  -20.49    456   1.01
    lp_parts[176]  -76.76    4.58   97.71   -366.5   -75.74  -42.57   -25.99  -20.48    455   1.01
    lp_parts[177]  -77.25    4.62   98.66   -369.8   -76.63  -42.43   -25.96  -20.48    455   1.01
    lp_parts[178]  -77.75    4.67   99.61   -373.1   -77.54  -42.64   -25.93  -20.46    455   1.01
    lp_parts[179]  -78.25    4.72  100.57   -376.4   -78.46  -42.63   -25.93  -20.45    455   1.01
    lp_parts[180]  -78.76    4.76  101.53   -379.7   -79.22  -42.71   -25.94  -20.45    454   1.01
    lp_parts[181]  -79.27    4.81  102.49   -383.0   -78.76  -42.64   -25.97  -20.45    454   1.01
    lp_parts[182]  -79.78    4.85  103.45   -386.3   -78.31  -42.49   -25.97  -20.46    454   1.01
    lp_parts[183]   -80.3     4.9  104.41   -389.6    -78.6  -42.55   -25.91  -20.46    454    1.0
    lp_parts[184]  -80.82    4.95  105.37   -392.9    -79.5  -42.61   -25.99  -20.46    454    1.0
    lp_parts[185]  -81.34    4.99  106.33   -396.2    -80.4  -42.51   -25.94  -20.46    453    1.0
    lp_parts[186]  -81.87    5.04   107.3   -399.6   -81.31  -42.47   -25.82  -20.45    453    1.0
    lp_parts[187]   -82.4    5.09  108.26   -402.9   -82.22  -42.34   -25.77  -20.45    453    1.0
    lp_parts[188]  -82.94    5.13  109.23   -406.2   -83.13  -42.21   -25.76  -20.45    453    1.0
    lp_parts[189]  -83.48    5.18   110.2   -409.6   -84.04  -42.31   -25.72  -20.46    453    1.0
    lp_parts[190]  -84.02    5.23  111.17   -412.9   -84.96   -42.5   -25.87  -20.46    453    1.0
    lp_parts[191]  -84.57    5.27  112.14   -416.2   -85.89  -42.55   -25.89  -20.46    452    1.0
    lp_parts[192]  -85.11    5.32  113.11   -419.6   -86.81  -42.65   -26.02  -20.46    452    1.0
    lp_parts[193]  -85.67    5.37  114.08   -422.9   -87.74  -42.75   -25.94  -20.46    452    1.0
    lp_parts[194]  -86.22    5.41  115.06   -426.3   -88.67  -42.52   -25.95  -20.46    452    1.0
    lp_parts[195]  -86.78    5.46  116.03   -429.6   -89.61  -42.44   -26.03  -20.46    452    1.0
    lp_parts[196]  -87.34    5.51  117.01   -433.0   -90.55  -42.35   -26.09  -20.46    451    1.0
    lp_parts[197]  -87.91    5.55  117.99   -436.3   -91.49  -42.35   -26.18  -20.46    451    1.0
    lp_parts[198]  -88.47     5.6  118.96   -439.7   -92.43  -42.45   -26.21  -20.47    451    1.0
    lp_parts[199]  -89.05    5.65  119.94   -443.0   -93.38  -42.75   -26.12  -20.47    451    1.0
    lp_parts[200]  -89.62     5.7  120.92   -446.4   -94.33   -42.7   -26.13  -20.46    451    1.0
    lp_parts[201]   -90.2    5.74   121.9   -449.8   -95.29  -42.45   -26.19  -20.47    451    1.0
    lp_parts[202]  -90.78    5.79  122.88   -453.2   -96.25  -42.42   -26.15  -20.48    450    1.0
    lp_parts[203]  -91.36    5.84  123.86   -456.5   -97.21  -42.34   -26.13  -20.47    450    1.0
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    lp_parts[207]  -93.72    6.03   127.8   -470.1   -101.0  -42.52   -26.09  -20.47    450    1.0
    lp_parts[208]  -94.32    6.07  128.78   -473.5   -102.0  -42.26   -26.18  -20.46    450    1.0
    lp_parts[209]  -94.92    6.12  129.77   -476.8   -103.0  -42.29    -26.2  -20.47    449    1.0
    lp_parts[210]  -95.52    6.17  130.75   -480.2   -104.0  -42.14   -26.09  -20.47    449    1.0
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    lp_parts[212]  -96.74    6.26  132.73   -487.0   -105.9  -41.99   -26.02  -20.46    449    1.0
    lp_parts[213]  -97.35    6.31  133.72   -490.4   -106.9  -42.15   -25.92  -20.45    449    1.0
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    lp_parts[355]  -202.7   13.24  277.14   -992.4   -265.8  -85.79   -26.58  -20.48    438    1.0
    lp_parts[356]  -203.5   13.29  278.17   -996.0   -267.1  -86.27   -26.56  -20.48    438    1.0
    lp_parts[357]  -204.4   13.34  279.19   -999.7   -268.3  -86.75   -26.56  -20.48    438    1.0
    lp_parts[358]  -205.2   13.39  280.21    -1003   -269.5  -87.24    -26.6  -20.48    438    1.0
    lp_parts[359]  -206.1   13.44  281.23    -1006   -270.7  -87.72   -26.69  -20.48    438    1.0
    lp_parts[360]  -206.9   13.49  282.25    -1010   -271.9  -88.21   -26.73  -20.48    438    1.0
    lp_parts[361]  -207.7   13.54  283.27    -1014   -273.1   -88.7   -26.85  -20.48    438    1.0
    lp_parts[362]  -208.6   13.59  284.29    -1017   -274.3  -89.19   -26.97  -20.47    438    1.0
    lp_parts[363]  -209.4   13.63  285.31    -1021   -275.6  -89.67   -27.04  -20.47    438    1.0
    lp_parts[364]  -210.2   13.68  286.33    -1025   -276.8  -90.16   -26.95  -20.47    438    1.0
    lp_parts[365]  -211.1   13.73  287.35    -1028   -278.0  -90.66    -27.0  -20.47    438    1.0
    lp_parts[366]  -211.9   13.78  288.37    -1032   -279.2  -91.15   -27.12  -20.47    438    1.0
    lp_parts[367]  -212.8   13.83  289.39    -1036   -280.4  -91.64   -27.24  -20.47    438    1.0
    lp_parts[368]  -213.6   13.88  290.41    -1039   -281.7  -92.13   -27.36  -20.48    438    1.0
    lp_parts[369]  -214.5   13.93  291.44    -1043   -282.9  -92.63   -27.49  -20.48    438    1.0
    lp_parts[370]  -215.3   13.98  292.46    -1046   -284.1  -93.12   -27.61  -20.48    438    1.0
    lp_parts[371]  -216.2   14.03  293.48    -1050   -285.3  -93.62   -27.66  -20.48    438    1.0
    lp_parts[372]  -217.0   14.08   294.5    -1054   -286.5  -94.11   -27.73  -20.49    438    1.0
    lp_parts[373]  -217.9   14.13  295.52    -1057   -287.8  -94.61   -27.79  -20.49    438    1.0
    lp_parts[374]  -218.7   14.18  296.54    -1061   -289.0  -95.11   -27.92  -20.48    437    1.0
    lp_parts[375]  -219.6   14.23  297.56    -1065   -290.2  -95.61   -28.04  -20.48    437    1.0
    lp_parts[376]  -220.4   14.28  298.58    -1068   -291.5  -96.11   -28.17  -20.48    437    1.0
    lp_parts[377]  -221.3   14.33  299.61    -1072   -292.7  -96.61    -28.3  -20.47    437    1.0
    lp_parts[378]  -222.1   14.38  300.63    -1076   -293.9  -97.11   -28.42  -20.47    437    1.0
    lp_parts[379]  -223.0   14.42  301.65    -1079   -295.1  -97.61    -28.5  -20.48    437    1.0
    lp_parts[380]  -223.8   14.47  302.67    -1083   -296.4  -98.12   -28.58  -20.48    437    1.0
    lp_parts[381]  -224.7   14.52  303.69    -1087   -297.6  -98.62   -28.71  -20.48    437    1.0
    lp_parts[382]  -225.5   14.57  304.71    -1090   -298.8  -99.12   -28.84  -20.48    437    1.0
    lp_parts[383]  -226.4   14.62  305.74    -1094   -300.1  -99.63   -28.97  -20.49    437    1.0
    lp_parts[384]  -227.2   14.67  306.76    -1097   -301.3  -100.1    -29.1   -20.5    437    1.0
    lp_parts[385]  -228.1   14.72  307.78    -1101   -302.5  -100.6   -29.23   -20.5    437    1.0
    lp_parts[386]  -229.0   14.77   308.8    -1105   -303.8  -101.1   -29.37   -20.5    437    1.0
    lp_parts[387]  -229.8   14.82  309.82    -1108   -305.0  -101.6    -29.5  -20.49    437    1.0
    lp_parts[388]  -230.7   14.87  310.85    -1112   -306.2  -102.1   -29.64  -20.48    437    1.0
    lp_parts[389]  -231.5   14.92  311.87    -1116   -307.5  -102.6   -29.77  -20.48    437    1.0
    lp_parts[390]  -232.4   14.97  312.89    -1119   -308.7  -103.1   -29.91  -20.48    437    1.0
    lp_parts[391]  -233.3   15.02  313.91    -1123   -310.0  -103.7   -29.94  -20.48    437    1.0
    lp_parts[392]  -234.1   15.07  314.94    -1127   -311.2  -104.2   -29.83  -20.48    437    1.0
    lp_parts[393]  -235.0   15.12  315.96    -1130   -312.4  -104.7   -29.74  -20.48    437    1.0
    lp_parts[394]  -235.9   15.17  316.98    -1134   -313.7  -105.2   -29.83  -20.47    437    1.0
    lp_parts[395]  -236.7   15.22   318.0    -1138   -314.9  -105.7   -29.96  -20.48    437    1.0
    lp_parts[396]  -237.6   15.26  319.03    -1141   -316.2  -106.2    -30.1  -20.48    437    1.0
    lp_parts[397]  -238.5   15.31  320.05    -1145   -317.4  -106.7   -30.23  -20.49    437    1.0
    lp_parts[398]  -239.3   15.36  321.07    -1149   -318.7  -107.3   -30.37  -20.49    437    1.0
    lp_parts[399]  -240.2   15.41  322.09    -1152   -319.9  -107.8   -30.51   -20.5    437    1.0
    lp_parts[400]  -241.1   15.46  323.12    -1156   -321.2  -108.3   -30.65  -20.49    437    1.0
    lp_parts[401]  -241.9   15.51  324.14    -1160   -322.4  -108.8   -30.79  -20.48    437    1.0
    lp_parts[402]  -242.8   15.56  325.16    -1163   -323.6  -109.3   -30.93  -20.48    437    1.0
    lp_parts[403]  -243.7   15.61  326.19    -1167   -324.9  -109.9   -31.07  -20.49    437    1.0
    lp_parts[404]  -244.5   15.66  327.21    -1171   -326.1  -110.4   -31.21  -20.48    437    1.0
    lp_parts[405]  -245.4   15.71  328.23    -1174   -327.4  -110.9   -31.35  -20.48    436    1.0
    lp_parts[406]  -246.3   15.76  329.25    -1178   -328.6  -111.4   -31.49  -20.47    436    1.0
    lp_parts[407]  -247.2   15.81  330.28    -1182   -329.9  -111.9   -31.64  -20.47    436    1.0
    lp_parts[408]  -248.0   15.86   331.3    -1185   -331.1  -112.5   -31.78  -20.47    436    1.0
    lp_parts[409]  -248.9   15.91  332.32    -1189   -332.4  -113.0   -31.92  -20.47    436    1.0
    lp_parts[410]  -249.8   15.96  333.35    -1193   -333.7  -113.5   -32.07  -20.48    436    1.0
    lp_parts[411]  -250.7   16.01  334.37    -1196   -334.9  -114.0   -32.22  -20.47    436    1.0
    lp_parts[412]  -251.5   16.06  335.39    -1200   -336.2  -114.6   -32.36  -20.47    436    1.0
    lp_parts[413]  -252.4   16.11  336.42    -1204   -337.4  -115.1   -32.51  -20.47    436    1.0
    lp_parts[414]  -253.3   16.16  337.44    -1207   -338.7  -115.6   -32.66  -20.47    436    1.0
    lp_parts[415]  -254.2   16.21  338.46    -1211   -339.9  -116.2   -32.81  -20.46    436    1.0
    lp_parts[416]  -255.1   16.26  339.49    -1215   -341.2  -116.7   -32.96  -20.46    436    1.0
    lp_parts[417]  -255.9   16.31  340.51    -1218   -342.4  -117.2   -33.11  -20.46    436    1.0
    lp_parts[418]  -256.8   16.36  341.53    -1222   -343.7  -117.7   -33.26  -20.46    436    1.0
    lp_parts[419]  -257.7    16.4  342.56    -1226   -345.0  -118.3   -33.41  -20.46    436    1.0
    lp_parts[420]  -258.6   16.45  343.58    -1229   -346.2  -118.8   -33.56  -20.47    436    1.0
    lp_parts[421]  -259.5    16.5   344.6    -1233   -347.5  -119.3   -33.71  -20.47    436    1.0
    lp_parts[422]  -260.4   16.55  345.63    -1237   -348.7  -119.9   -33.87  -20.47    436    1.0
    lp_parts[423]  -261.2    16.6  346.65    -1240   -350.0  -120.4   -34.02  -20.48    436    1.0
    lp_parts[424]  -262.1   16.65  347.67    -1244   -351.3  -120.9   -34.18  -20.47    436    1.0
    lp_parts[425]  -263.0    16.7   348.7    -1248   -352.5  -121.5   -34.33  -20.47    436    1.0
    lp_parts[426]  -263.9   16.75  349.72    -1251   -353.8  -122.0   -34.49  -20.47    436    1.0
    lp_parts[427]  -264.8    16.8  350.75    -1255   -355.1  -122.5   -34.64  -20.47    436    1.0
    lp_parts[428]  -265.7   16.85  351.77    -1259   -356.3  -123.1    -34.8  -20.47    436    1.0
    lp_parts[429]  -266.6    16.9  352.79    -1262   -357.6  -123.6   -34.96  -20.48    436    1.0
    lp_parts[430]  -267.4   16.95  353.82    -1266   -358.9  -124.2   -35.12  -20.48    436    1.0
    lp_parts[431]  -268.3    17.0  354.84    -1270   -360.1  -124.7   -35.28  -20.49    436    1.0
    lp_parts[432]  -269.2   17.05  355.86    -1274   -361.4  -125.2   -35.44  -20.49    436    1.0
    lp_parts[433]  -270.1    17.1  356.89    -1277   -362.7  -125.8    -35.6  -20.48    436    1.0
    lp_parts[434]  -271.0   17.15  357.91    -1281   -363.9  -126.3   -35.76  -20.49    436    1.0
    lp_parts[435]  -271.9    17.2  358.94    -1285   -365.2  -126.9   -35.92  -20.49    436    1.0
    lp_parts[436]  -272.8   17.25  359.96    -1288   -366.5  -127.4   -36.08  -20.49    436    1.0
    lp_parts[437]  -273.7    17.3  360.98    -1292   -367.7  -127.9   -36.24   -20.5    436    1.0
    lp_parts[438]  -274.6   17.35  362.01    -1296   -369.0  -128.5   -36.41   -20.5    435    1.0
    lp_parts[439]  -275.5    17.4  363.03    -1299   -370.3  -129.0   -36.57   -20.5    435    1.0
    lp_parts[440]  -276.4   17.45  364.06    -1303   -371.5  -129.6   -36.74   -20.5    435    1.0
    lp_parts[441]  -277.3    17.5  365.08    -1307   -372.8  -130.1    -36.9  -20.49    435    1.0
    lp_parts[442]  -278.2   17.55  366.11    -1310   -374.1  -130.7   -37.07  -20.49    435    1.0
    lp_parts[443]  -279.1    17.6  367.13    -1314   -375.4  -131.2   -37.23  -20.48    435    1.0
    lp_parts[444]  -280.0   17.65  368.15    -1318   -376.6  -131.8    -37.4  -20.48    435    1.0
    lp_parts[445]  -280.9   17.69  369.18    -1321   -377.9  -132.3   -37.57  -20.48    435    1.0
    lp_parts[446]  -281.8   17.74   370.2    -1325   -379.2  -132.8   -37.73  -20.49    435    1.0
    lp_parts[447]  -282.7   17.79  371.23    -1329   -380.5  -133.4    -37.9   -20.5    435    1.0
    lp_parts[448]  -283.6   17.84  372.25    -1333   -381.7  -133.9   -38.07  -20.49    435    1.0
    lp_parts[449]  -284.5   17.89  373.28    -1336   -383.0  -134.5   -38.24   -20.5    435    1.0
    lp_parts[450]  -285.4   17.94   374.3    -1340   -384.3  -135.0   -38.41  -20.51    435    1.0
    lp_parts[451]  -286.3   17.99  375.32    -1344   -385.6  -135.6   -38.58  -20.52    435    1.0
    lp_parts[452]  -287.2   18.04  376.35    -1347   -386.9  -136.1   -38.75  -20.52    435    1.0
    lp_parts[453]  -288.1   18.09  377.37    -1351   -388.1  -136.7   -38.93  -20.52    435    1.0
    lp_parts[454]  -289.0   18.14   378.4    -1355   -389.4  -137.2    -39.1  -20.53    435    1.0
    lp_parts[455]  -289.9   18.19  379.42    -1358   -390.7  -137.8   -39.27  -20.53    435    1.0
    lp_parts[456]  -290.8   18.24  380.45    -1362   -392.0  -138.4   -39.44  -20.53    435    1.0
    lp_parts[457]  -291.7   18.29  381.47    -1366   -393.3  -138.9   -39.62  -20.52    435    1.0
    lp_parts[458]  -292.6   18.34   382.5    -1369   -394.5  -139.5   -39.79  -20.53    435    1.0
    lp_parts[459]  -293.5   18.39  383.52    -1373   -395.8  -140.0   -39.97  -20.54    435    1.0
    lp_parts[460]  -294.4   18.44  384.55    -1377   -397.1  -140.6   -40.14  -20.53    435    1.0
    lp_parts[461]  -295.3   18.49  385.57    -1381   -398.4  -141.1   -40.32  -20.52    435    1.0
    lp_parts[462]  -296.2   18.54   386.6    -1384   -399.7  -141.7    -40.5  -20.52    435    1.0
    lp_parts[463]  -297.1   18.59  387.62    -1388   -401.0  -142.2   -40.67  -20.53    435    1.0
    lp_parts[464]  -298.0   18.64  388.65    -1392   -402.2  -142.8   -40.85  -20.54    435    1.0
    lp_parts[465]  -298.9   18.69  389.67    -1395   -403.5  -143.4   -41.03  -20.55    435    1.0
    lp_parts[466]  -299.8   18.74   390.7    -1399   -404.8  -143.9   -41.21  -20.55    435    1.0
    lp_parts[467]  -300.8   18.79  391.72    -1403   -406.1  -144.5   -41.39  -20.55    435    1.0
    lp_parts[468]  -301.7   18.84  392.75    -1406   -407.4  -145.0   -41.57  -20.55    435    1.0
    lp_parts[469]  -302.6   18.89  393.77    -1410   -408.7  -145.6   -41.75  -20.56    435    1.0
    lp_parts[470]  -303.5   18.94   394.8    -1414   -410.0  -146.2   -41.93  -20.57    435    1.0
    lp_parts[471]  -304.4   18.99  395.82    -1418   -411.3  -146.7   -42.11  -20.56    435    1.0
    lp_parts[472]  -305.3   19.04  396.85    -1421   -412.5  -147.3   -42.29  -20.57    435    1.0
    lp_parts[473]  -306.2   19.09  397.87    -1425   -413.8  -147.8   -42.48  -20.58    435    1.0
    lp_parts[474]  -307.1   19.14   398.9    -1429   -415.1  -148.4   -42.66  -20.58    435    1.0
    lp_parts[475]  -308.1   19.18  399.92    -1432   -416.4  -149.0   -42.84  -20.59    435    1.0
    lp_parts[476]  -309.0   19.23  400.95    -1436   -417.7  -149.5   -43.03   -20.6    435    1.0
    lp_parts[477]  -309.9   19.28  401.97    -1440   -419.0  -150.1   -43.21   -20.6    435    1.0
    lp_parts[478]  -310.8   19.33   403.0    -1444   -420.3  -150.7   -43.39  -20.59    434    1.0
    lp_parts[479]  -311.7   19.38  404.02    -1447   -421.6  -151.2   -43.58  -20.59    434    1.0
    lp_parts[480]  -312.6   19.43  405.05    -1451   -422.9  -151.8   -43.77   -20.6    434    1.0
    lp_parts[481]  -313.5   19.48  406.07    -1455   -424.2  -152.4   -43.95  -20.61    434    1.0
    lp_parts[482]  -314.5   19.53   407.1    -1458   -425.5  -152.9   -44.14  -20.62    434    1.0
    lp_parts[483]  -315.4   19.58  408.12    -1462   -426.8  -153.5   -44.33  -20.62    434    1.0
    lp_parts[484]  -316.3   19.63  409.15    -1466   -428.1  -154.1   -44.52  -20.64    434    1.0
    lp_parts[485]  -317.2   19.68  410.17    -1470   -429.4  -154.6    -44.7  -20.65    434    1.0
    lp_parts[486]  -318.1   19.73   411.2    -1473   -430.7  -155.2   -44.89  -20.67    434    1.0
    lp_parts[487]  -319.1   19.78  412.22    -1477   -431.9  -155.8   -45.08  -20.68    434    1.0
    lp_parts[488]  -320.0   19.83  413.25    -1481   -433.2  -156.3   -45.27  -20.69    434    1.0
    lp_parts[489]  -320.9   19.88  414.27    -1484   -434.5  -156.9   -45.46  -20.71    434    1.0
    lp_parts[490]  -321.8   19.93   415.3    -1488   -435.8  -157.5   -45.65  -20.73    434    1.0
    lp_parts[491]  -322.7   19.98  416.33    -1492   -437.1  -158.0   -45.84  -20.74    434    1.0
    lp_parts[492]  -323.7   20.03  417.35    -1496   -438.4  -158.6   -46.04  -20.76    434    1.0
    lp_parts[493]  -324.6   20.08  418.38    -1499   -439.7  -159.2   -46.23  -20.78    434    1.0
    lp_parts[494]  -325.5   20.13   419.4    -1503   -441.0  -159.8   -46.42   -20.8    434    1.0
    lp_parts[495]  -326.4   20.18  420.43    -1507   -442.3  -160.3   -46.61  -20.82    434    1.0
    lp_parts[496]  -327.4   20.23  421.45    -1510   -443.6  -160.9   -46.81  -20.84    434    1.0
    lp_parts[497]  -328.3   20.28  422.48    -1514   -444.9  -161.5    -47.0  -20.85    434    1.0
    lp_parts[498]  -329.2   20.33   423.5    -1518   -446.2  -162.1    -47.2  -20.85    434    1.0
    lp_parts[499]  -330.1   20.38  424.53    -1522   -447.5  -162.6   -47.39  -20.86    434    1.0
    lp_parts[500]  -331.1   20.43  425.56    -1525   -448.9  -163.2   -47.59  -20.86    434    1.0
    n              186.53    6.66  125.54     39.0     83.0   147.0   267.55   463.6    356    1.0
    prob_n[1]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[2]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[3]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[4]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[5]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[6]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[7]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[8]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[9]         0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[10]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[11]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[12]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[13]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[14]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[15]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[16]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[17]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[18]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[19]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[20]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[21]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[22]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[23]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[24]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[25]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[26]        0.0     nan     0.0      0.0      0.0     0.0      0.0     0.0    nan    nan
    prob_n[27]     1.2e-4  1.1e-4  3.2e-3 2.8e-120  1.3e-94 1.4e-66  7.9e-41 1.3e-11    844    1.0
    prob_n[28]     5.6e-4  4.7e-4    0.01 1.8e-116  6.8e-91 5.1e-63  1.4e-37  2.0e-9    529   1.01
    prob_n[29]     1.1e-3  8.4e-4    0.02 3.6e-113  1.2e-87 5.8e-60  7.7e-35 10.0e-8    387   1.01
    prob_n[30]     1.4e-3  9.4e-4    0.02 3.4e-110  9.2e-85 3.2e-57  2.0e-32  2.3e-6    354   1.01
    prob_n[31]     1.4e-3  8.1e-4    0.02 1.9e-107  4.1e-82 9.7e-55  3.0e-30  3.0e-5    414   1.01
    prob_n[32]     1.4e-3  6.2e-4    0.02 6.3e-105  1.2e-79 1.9e-52  2.8e-28  2.5e-4    630   1.01
    prob_n[33]     1.5e-3  4.9e-4    0.01 1.4e-102  2.2e-77 2.5e-50  1.8e-26  1.0e-3    909    1.0
    prob_n[34]     1.8e-3  5.1e-4    0.01 2.4e-100  3.1e-75 2.3e-48  8.0e-25  2.5e-3    833    1.0
    prob_n[35]     2.2e-3  6.3e-4    0.02  2.9e-98  3.2e-73 1.6e-46  2.7e-23  6.6e-3    678    1.0
    prob_n[36]     2.6e-3  7.6e-4    0.02  2.8e-96  2.5e-71 8.9e-45  7.2e-22    0.02    604    1.0
    prob_n[37]     3.1e-3  8.3e-4    0.02  2.1e-94  1.6e-69 3.9e-43  1.5e-20    0.02    593    1.0
    prob_n[38]     3.3e-3  8.3e-4    0.02  1.3e-92  8.3e-68 1.4e-41  2.6e-19    0.04    617    1.0
    prob_n[39]     3.5e-3  7.5e-4    0.02  6.9e-91  3.6e-66 4.1e-40  3.7e-18    0.06    698    1.0
    prob_n[40]     3.6e-3  6.5e-4    0.02  3.0e-89  1.3e-64 1.0e-38  4.5e-17    0.07    862    1.0
    prob_n[41]     3.7e-3  5.9e-4    0.02  1.1e-87  4.1e-63 2.2e-37  4.6e-16    0.07   1036    1.0
    prob_n[42]     3.9e-3  5.8e-4    0.02  3.7e-86  1.1e-61 4.0e-36  4.0e-15    0.07   1105    1.0
    prob_n[43]     4.0e-3  5.9e-4    0.02  1.0e-84  2.6e-60 6.5e-35  2.8e-14    0.07   1102    1.0
    prob_n[44]     4.2e-3  5.9e-4    0.02  2.6e-83  5.5e-59 9.4e-34  1.7e-13    0.08   1086    1.0
    prob_n[45]     4.4e-3  5.9e-4    0.02  5.9e-82  1.0e-57 1.2e-32  8.1e-13    0.09   1089    1.0
    prob_n[46]     4.6e-3  5.8e-4    0.02  1.2e-80  1.8e-56 1.4e-31  4.3e-12    0.09   1114    1.0
    prob_n[47]     4.7e-3  5.7e-4    0.02  2.2e-79  2.7e-55 1.5e-30  1.5e-11    0.08   1146    1.0
    prob_n[48]     4.9e-3  5.7e-4    0.02  3.7e-78  3.7e-54 1.3e-29  7.2e-11    0.09   1170    1.0
    prob_n[49]     5.2e-3  5.7e-4    0.02  5.6e-77  4.8e-53 1.2e-28  2.6e-10    0.09   1155    1.0
    prob_n[50]     5.4e-3  5.7e-4    0.02  7.9e-76  5.6e-52 8.9e-28  9.0e-10    0.08   1175    1.0
    prob_n[51]     5.6e-3  5.8e-4    0.02  1.0e-74  6.1e-51 5.9e-27   3.3e-9    0.09   1151    1.0
    prob_n[52]     5.9e-3  6.0e-4    0.02  1.3e-73  6.2e-50 3.9e-26   7.6e-9    0.09   1124    1.0
    prob_n[53]     6.1e-3  6.3e-4    0.02  1.4e-72  5.8e-49 2.4e-25   1.4e-8    0.09   1025    1.0
    prob_n[54]     6.2e-3  6.6e-4    0.02  1.5e-71  5.2e-48 1.4e-24   2.4e-8    0.09    950    1.0
    prob_n[55]     6.4e-3  6.8e-4    0.02  1.5e-70  4.3e-47 7.2e-24   6.0e-8    0.09    892    1.0
    prob_n[56]     6.4e-3  6.9e-4    0.02  1.4e-69  3.1e-46 3.1e-23   1.4e-7    0.09    867    1.0
    prob_n[57]     6.4e-3  6.6e-4    0.02  1.2e-68  2.3e-45 1.6e-22   2.9e-7    0.08    905    1.0
    prob_n[58]     6.4e-3  6.3e-4    0.02  1.0e-67  1.5e-44 7.1e-22   4.4e-7    0.08    960    1.0
    prob_n[59]     6.3e-3  6.0e-4    0.02  8.3e-67  6.2e-44 2.2e-21   7.3e-7    0.08   1019    1.0
    prob_n[60]     6.1e-3  5.8e-4    0.02  6.3e-66  3.3e-43 6.4e-21   1.0e-6    0.08   1026    1.0
    prob_n[61]     6.0e-3  5.6e-4    0.02  4.5e-65  1.8e-42 2.7e-20   1.6e-6    0.08   1043    1.0
    prob_n[62]     5.9e-3  5.4e-4    0.02  3.1e-64  1.0e-41 1.0e-19   2.8e-6    0.08   1067    1.0
    prob_n[63]     5.7e-3  5.2e-4    0.02  1.6e-63  5.0e-41 2.9e-19   3.8e-6    0.07   1095    1.0
    prob_n[64]     5.6e-3  5.0e-4    0.02  1.0e-62  2.2e-40 7.4e-19   5.1e-6    0.07   1123    1.0
    prob_n[65]     5.5e-3  4.9e-4    0.02  6.4e-62  1.1e-39 1.3e-18   5.8e-6    0.07   1111    1.0
    prob_n[66]     5.4e-3  4.8e-4    0.02  2.8e-61  3.7e-39 2.7e-18   8.9e-6    0.07   1091    1.0
    prob_n[67]     5.3e-3  4.8e-4    0.02  1.3e-60  1.7e-38 5.9e-18   1.2e-5    0.07   1060    1.0
    prob_n[68]     5.3e-3  4.9e-4    0.02  6.6e-60  8.0e-38 1.8e-17   1.4e-5    0.07   1019    1.0
    prob_n[69]     5.2e-3  4.9e-4    0.02  3.5e-59  3.3e-37 3.4e-17   1.5e-5    0.07    975    1.0
    prob_n[70]     5.2e-3  5.0e-4    0.02  1.8e-58  1.1e-36 7.9e-17   2.1e-5    0.07    942    1.0
    prob_n[71]     5.2e-3  5.0e-4    0.02  7.9e-58  4.1e-36 1.3e-16   2.2e-5    0.06    910    1.0
    prob_n[72]     5.2e-3  5.0e-4    0.02  3.8e-57  1.4e-35 2.4e-16   2.2e-5    0.06    894    1.0
    prob_n[73]     5.2e-3  5.0e-4    0.01  1.6e-56  5.3e-35 3.6e-16   2.1e-5    0.06    885    1.0
    prob_n[74]     5.2e-3  5.0e-4    0.01  7.2e-56  2.0e-34 4.6e-16   2.6e-5    0.06    886    1.0
    prob_n[75]     5.2e-3  4.9e-4    0.01  1.8e-55  4.4e-34 9.3e-16   2.8e-5    0.06    891    1.0
    prob_n[76]     5.1e-3  4.8e-4    0.01  4.0e-55  1.0e-33 1.8e-15   3.4e-5    0.06    899    1.0
    prob_n[77]     5.1e-3  4.7e-4    0.01  1.7e-54  3.3e-33 2.8e-15   3.2e-5    0.06    919    1.0
    prob_n[78]     5.1e-3  4.6e-4    0.01  6.8e-54  3.9e-33 4.5e-15   3.4e-5    0.06    952    1.0
    prob_n[79]     5.1e-3  4.5e-4    0.01  1.4e-53  1.2e-32 9.2e-15   3.4e-5    0.06    995    1.0
    prob_n[80]     5.1e-3  4.3e-4    0.01  4.5e-53  2.8e-32 1.3e-14   4.1e-5    0.06   1047    1.0
    prob_n[81]     5.0e-3  4.2e-4    0.01  1.4e-52  7.5e-32 1.4e-14   4.7e-5    0.06   1104    1.0
    prob_n[82]     5.0e-3  4.0e-4    0.01  4.7e-52  2.1e-31 1.8e-14   5.1e-5    0.06   1157    1.0
    prob_n[83]     5.0e-3  3.9e-4    0.01  1.0e-51  4.0e-31 1.4e-14   4.8e-5    0.05   1192    1.0
    prob_n[84]     4.9e-3  3.8e-4    0.01  3.6e-51  6.0e-31 2.0e-14   5.7e-5    0.05   1222    1.0
    prob_n[85]     4.9e-3  3.7e-4    0.01  1.2e-50  1.5e-30 2.7e-14   6.4e-5    0.05   1246    1.0
    prob_n[86]     4.8e-3  3.7e-4    0.01  3.6e-50  3.3e-30 4.9e-14   6.2e-5    0.05   1265    1.0
    prob_n[87]     4.8e-3  3.6e-4    0.01  1.0e-49  8.6e-30 7.6e-14   8.0e-5    0.05   1280    1.0
    prob_n[88]     4.7e-3  3.5e-4    0.01  1.2e-49  1.6e-29 1.3e-13   8.2e-5    0.05   1293    1.0
    prob_n[89]     4.7e-3  3.5e-4    0.01  2.1e-49  3.3e-29 1.2e-13   9.1e-5    0.05   1306    1.0
    prob_n[90]     4.6e-3  3.4e-4    0.01  3.7e-49  3.9e-29 1.3e-13   8.7e-5    0.05   1321    1.0
    prob_n[91]     4.6e-3  3.3e-4    0.01  3.1e-49  3.9e-29 1.8e-13   9.4e-5    0.05   1339    1.0
    prob_n[92]     4.5e-3  3.3e-4    0.01  1.4e-49  6.5e-29 1.9e-13   1.2e-4    0.05   1360    1.0
    prob_n[93]     4.5e-3  3.2e-4    0.01  1.3e-49  1.4e-28 2.3e-13   1.2e-4    0.05   1384    1.0
    prob_n[94]     4.5e-3  3.1e-4    0.01  1.5e-50  2.1e-28 2.6e-13   9.8e-5    0.05   1408    1.0
    prob_n[95]     4.4e-3  3.0e-4    0.01  8.9e-52  3.3e-28 3.2e-13   1.1e-4    0.05   1451    1.0
    prob_n[96]     4.4e-3  2.9e-4    0.01  2.3e-52  7.1e-28 4.1e-13   1.1e-4    0.05   1507    1.0
    prob_n[97]     4.3e-3  2.9e-4    0.01  6.7e-52  1.4e-27 4.1e-13   1.1e-4    0.05   1554    1.0
    prob_n[98]     4.3e-3  2.8e-4    0.01  2.0e-51  1.6e-27 4.2e-13   1.1e-4    0.05   1557    1.0
    prob_n[99]     4.3e-3  2.8e-4    0.01  5.6e-51  1.9e-27 4.5e-13   1.1e-4    0.04   1559    1.0
    prob_n[100]    4.3e-3  2.8e-4    0.01  7.9e-51  2.4e-27 6.4e-13   1.3e-4    0.04   1560    1.0
    prob_n[101]    4.2e-3  2.8e-4    0.01  3.8e-51  3.2e-27 7.6e-13   1.2e-4    0.04   1561    1.0
    prob_n[102]    4.2e-3  2.8e-4    0.01  2.6e-52  3.5e-27 6.4e-13   1.4e-4    0.04   1560    1.0
    prob_n[103]    4.2e-3  2.7e-4    0.01  1.7e-53  5.9e-27 7.5e-13   1.5e-4    0.04   1559    1.0
    prob_n[104]    4.2e-3  2.7e-4    0.01  1.1e-54  8.0e-27 5.8e-13   1.5e-4    0.04   1557    1.0
    prob_n[105]    4.2e-3  2.7e-4    0.01  7.1e-56  5.4e-27 6.5e-13   1.5e-4    0.04   1555    1.0
    prob_n[106]    4.1e-3  2.7e-4    0.01  4.5e-57  1.1e-26 8.2e-13   1.3e-4    0.04   1552    1.0
    prob_n[107]    4.1e-3  2.7e-4    0.01  2.8e-58  1.2e-26 1.3e-12   1.1e-4    0.04   1550    1.0
    prob_n[108]    4.1e-3  2.7e-4    0.01  1.8e-59  2.3e-26 1.6e-12   1.4e-4    0.04   1548    1.0
    prob_n[109]    4.1e-3  2.7e-4    0.01  1.1e-60  3.3e-26 1.6e-12   1.5e-4    0.04   1546    1.0
    prob_n[110]    4.1e-3  2.6e-4    0.01  6.6e-62  3.9e-26 2.0e-12   1.7e-4    0.04   1544    1.0
    prob_n[111]    4.0e-3  2.6e-4    0.01  3.9e-63  2.7e-26 2.6e-12   1.6e-4    0.04   1544    1.0
    prob_n[112]    4.0e-3  2.6e-4    0.01  2.3e-64  4.0e-26 3.1e-12   1.7e-4    0.04   1543    1.0
    prob_n[113]    4.0e-3  2.6e-4    0.01  1.4e-65  4.6e-26 2.7e-12   1.7e-4    0.04   1520    1.0
    prob_n[114]    3.9e-3  2.6e-4 10.0e-3  8.0e-67  7.3e-26 2.5e-12   1.5e-4    0.04   1455    1.0
    prob_n[115]    3.9e-3  2.6e-4  9.9e-3  4.6e-68  7.1e-26 2.9e-12   1.5e-4    0.04   1394    1.0
    prob_n[116]    3.8e-3  2.7e-4  9.8e-3  2.6e-69  8.7e-26 4.1e-12   1.3e-4    0.04   1338    1.0
    prob_n[117]    3.8e-3  2.7e-4  9.6e-3  1.5e-70  9.6e-26 4.6e-12   1.5e-4    0.04   1286    1.0
    prob_n[118]    3.8e-3  2.7e-4  9.5e-3  8.2e-72  1.5e-25 4.5e-12   1.3e-4    0.04   1239    1.0
    prob_n[119]    3.7e-3  2.7e-4  9.4e-3  4.6e-73  1.4e-25 4.2e-12   1.3e-4    0.04   1199    1.0
    prob_n[120]    3.7e-3  2.7e-4  9.3e-3  2.5e-74  1.3e-25 4.1e-12   1.3e-4    0.04   1172    1.0
    prob_n[121]    3.6e-3  2.7e-4  9.2e-3  1.4e-75  1.7e-25 4.5e-12   1.3e-4    0.04   1149    1.0
    prob_n[122]    3.6e-3  2.7e-4  9.1e-3  7.3e-77  2.6e-25 3.4e-12   1.3e-4    0.04   1136    1.0
    prob_n[123]    3.6e-3  2.7e-4  9.0e-3  3.9e-78  3.0e-25 3.9e-12   1.3e-4    0.04   1137    1.0
    prob_n[124]    3.5e-3  2.6e-4  8.9e-3  2.1e-79  3.2e-25 3.5e-12   1.5e-4    0.04   1142    1.0
    prob_n[125]    3.5e-3  2.6e-4  8.8e-3  1.1e-80  3.0e-25 3.6e-12   1.4e-4    0.03   1149    1.0
    prob_n[126]    3.4e-3  2.5e-4  8.7e-3  5.7e-82  3.6e-25 4.4e-12   1.2e-4    0.03   1159    1.0
    prob_n[127]    3.4e-3  2.5e-4  8.6e-3  2.9e-83  3.2e-25 4.9e-12   1.0e-4    0.03   1176    1.0
    prob_n[128]    3.4e-3  2.5e-4  8.5e-3  1.5e-84  4.4e-25 5.2e-12   8.8e-5    0.03   1195    1.0
    prob_n[129]    3.3e-3  2.4e-4  8.4e-3  7.6e-86  5.7e-25 5.6e-12   8.7e-5    0.03   1214    1.0
    prob_n[130]    3.3e-3  2.4e-4  8.4e-3  3.8e-87  6.9e-25 6.3e-12   1.1e-4    0.03   1234    1.0
    prob_n[131]    3.3e-3  2.3e-4  8.3e-3  1.9e-88  7.4e-25 7.6e-12   1.2e-4    0.03   1253    1.0
    prob_n[132]    3.2e-3  2.3e-4  8.2e-3  9.5e-90  5.9e-25 9.2e-12   1.2e-4    0.03   1271    1.0
    prob_n[133]    3.2e-3  2.3e-4  8.2e-3  4.7e-91  8.5e-25 1.1e-11   1.1e-4    0.03   1288    1.0
    prob_n[134]    3.2e-3  2.3e-4  8.1e-3  2.3e-92  1.3e-24 1.0e-11   1.2e-4    0.03   1302    1.0
    prob_n[135]    3.2e-3  2.2e-4  8.1e-3  1.1e-93  1.5e-24 7.2e-12   1.2e-4    0.03   1314    1.0
    prob_n[136]    3.1e-3  2.2e-4  8.0e-3  5.4e-95  2.0e-24 5.4e-12   1.3e-4    0.03   1326    1.0
    prob_n[137]    3.1e-3  2.2e-4  8.0e-3  2.6e-96  1.9e-24 4.7e-12   1.3e-4    0.03   1337    1.0
    prob_n[138]    3.1e-3  2.2e-4  7.9e-3  1.2e-97  1.8e-24 5.8e-12   1.3e-4    0.03   1347    1.0
    prob_n[139]    3.1e-3  2.1e-4  7.9e-3  5.8e-99  2.6e-24 6.3e-12   1.2e-4    0.03   1357    1.0
    prob_n[140]    3.0e-3  2.1e-4  7.8e-3 2.7e-100  4.2e-24 6.0e-12   1.2e-4    0.03   1367    1.0
    prob_n[141]    3.0e-3  2.1e-4  7.8e-3 1.3e-101  4.2e-24 5.7e-12   1.2e-4    0.03   1377    1.0
    prob_n[142]    3.0e-3  2.1e-4  7.7e-3 5.9e-103  5.0e-24 5.7e-12   1.2e-4    0.03   1387    1.0
    prob_n[143]    2.9e-3  2.0e-4  7.7e-3 2.7e-104  7.4e-24 5.9e-12   1.3e-4    0.03   1397    1.0
    prob_n[144]    2.9e-3  2.0e-4  7.6e-3 1.2e-105  1.1e-23 6.7e-12   1.4e-4    0.03   1408    1.0
    prob_n[145]    2.9e-3  2.0e-4  7.5e-3 5.6e-107  1.1e-23 7.3e-12   1.3e-4    0.03   1418    1.0
    prob_n[146]    2.9e-3  2.0e-4  7.5e-3 2.5e-108  1.6e-23 6.7e-12   1.2e-4    0.03   1427    1.0
    prob_n[147]    2.8e-3  1.9e-4  7.4e-3 1.1e-109  1.4e-23 5.9e-12   1.2e-4    0.03   1437    1.0
    prob_n[148]    2.8e-3  1.9e-4  7.3e-3 5.1e-111  9.3e-24 5.2e-12   1.3e-4    0.03   1446    1.0
    prob_n[149]    2.8e-3  1.9e-4  7.2e-3 2.3e-112  1.5e-23 4.9e-12   1.2e-4    0.03   1457    1.0
    prob_n[150]    2.7e-3  1.9e-4  7.2e-3 1.0e-113  1.8e-23 6.1e-12   1.0e-4    0.03   1467    1.0
    prob_n[151]    2.7e-3  1.8e-4  7.1e-3 4.4e-115  2.2e-23 7.1e-12   1.1e-4    0.03   1478    1.0
    prob_n[152]    2.7e-3  1.8e-4  7.0e-3 1.9e-116  2.3e-23 5.9e-12  10.0e-5    0.03   1507    1.0
    prob_n[153]    2.7e-3  1.8e-4  6.9e-3 8.4e-118  1.9e-23 6.8e-12   9.6e-5    0.03   1542    1.0
    prob_n[154]    2.6e-3  1.7e-4  6.9e-3 3.6e-119  2.0e-23 5.1e-12   9.9e-5    0.03   1572    1.0
    prob_n[155]    2.6e-3  1.7e-4  6.8e-3 1.6e-120  2.4e-23 5.2e-12   9.9e-5    0.03   1597    1.0
    prob_n[156]    2.6e-3  1.7e-4  6.7e-3 6.6e-122  2.6e-23 4.3e-12   1.1e-4    0.03   1616    1.0
    prob_n[157]    2.6e-3  1.6e-4  6.7e-3 2.8e-123  2.8e-23 5.0e-12   1.2e-4    0.03   1638    1.0
    prob_n[158]    2.5e-3  1.6e-4  6.6e-3 1.2e-124  3.2e-23 5.9e-12   1.1e-4    0.03   1658    1.0
    prob_n[159]    2.5e-3  1.6e-4  6.5e-3 5.0e-126  2.8e-23 5.8e-12   1.1e-4    0.03   1675    1.0
    prob_n[160]    2.5e-3  1.6e-4  6.5e-3 2.1e-127  3.2e-23 6.0e-12   1.1e-4    0.03   1681    1.0
    prob_n[161]    2.5e-3  1.6e-4  6.4e-3 8.8e-129  2.0e-23 5.6e-12   1.1e-4    0.03   1672    1.0
    prob_n[162]    2.5e-3  1.6e-4  6.3e-3 3.7e-130  2.1e-23 4.3e-12   1.1e-4    0.03   1660    1.0
    prob_n[163]    2.4e-3  1.6e-4  6.3e-3 1.5e-131  2.5e-23 4.2e-12   1.1e-4    0.03   1644    1.0
    prob_n[164]    2.4e-3  1.5e-4  6.2e-3 6.2e-133  2.7e-23 3.5e-12   1.1e-4    0.03   1633    1.0
    prob_n[165]    2.4e-3  1.5e-4  6.2e-3 2.5e-134  1.8e-23 3.3e-12   1.1e-4    0.03   1625    1.0
    prob_n[166]    2.4e-3  1.5e-4  6.2e-3 1.0e-135  1.8e-23 3.5e-12   1.1e-4    0.03   1614    1.0
    prob_n[167]    2.4e-3  1.5e-4  6.1e-3 4.2e-137  1.3e-23 2.7e-12   1.1e-4    0.03   1603    1.0
    prob_n[168]    2.4e-3  1.5e-4  6.1e-3 1.7e-138  1.4e-23 3.7e-12   1.0e-4    0.02   1590    1.0
    prob_n[169]    2.4e-3  1.5e-4  6.1e-3 6.8e-140  7.2e-24 3.7e-12  10.0e-5    0.02   1576    1.0
    prob_n[170]    2.3e-3  1.5e-4  6.0e-3 2.7e-141  6.5e-24 3.8e-12   1.1e-4    0.02   1561    1.0
    prob_n[171]    2.3e-3  1.5e-4  6.0e-3 1.1e-142  3.3e-24 4.0e-12   1.2e-4    0.02   1546    1.0
    prob_n[172]    2.3e-3  1.5e-4  6.0e-3 4.3e-144  1.7e-24 3.9e-12   1.1e-4    0.02   1530    1.0
    prob_n[173]    2.3e-3  1.5e-4  6.0e-3 1.7e-145  6.8e-25 3.1e-12   1.0e-4    0.02   1515    1.0
    prob_n[174]    2.3e-3  1.5e-4  6.0e-3 6.7e-147  3.1e-25 3.6e-12   9.7e-5    0.02   1499    1.0
    prob_n[175]    2.3e-3  1.5e-4  5.9e-3 2.6e-148  1.3e-25 4.3e-12   8.8e-5    0.02   1484    1.0
    prob_n[176]    2.3e-3  1.5e-4  5.9e-3 1.0e-149  5.7e-26 4.7e-12   8.5e-5    0.02   1470    1.0
    prob_n[177]    2.3e-3  1.6e-4  5.9e-3 4.0e-151  2.3e-26 4.8e-12   8.6e-5    0.02   1457    1.0
    prob_n[178]    2.3e-3  1.6e-4  5.9e-3 1.5e-152  9.2e-27 5.0e-12   8.8e-5    0.02   1444    1.0
    prob_n[179]    2.3e-3  1.6e-4  5.9e-3 6.0e-154  6.6e-27 5.4e-12   9.6e-5    0.02   1433    1.0
    prob_n[180]    2.3e-3  1.6e-4  5.9e-3 2.3e-155  1.0e-26 4.4e-12   9.5e-5    0.02   1422    1.0
    prob_n[181]    2.3e-3  1.6e-4  5.9e-3 8.8e-157  1.6e-26 4.4e-12   9.4e-5    0.02   1413    1.0
    prob_n[182]    2.3e-3  1.6e-4  5.9e-3 3.4e-158  7.6e-27 4.3e-12   8.6e-5    0.02   1405    1.0
    prob_n[183]    2.3e-3  1.6e-4  5.9e-3 1.3e-159  3.1e-27 4.7e-12   9.2e-5    0.02   1398    1.0
    prob_n[184]    2.3e-3  1.6e-4  5.9e-3 4.8e-161  1.3e-27 5.1e-12   8.9e-5    0.02   1393    1.0
    prob_n[185]    2.3e-3  1.6e-4  5.8e-3 1.8e-162  5.1e-28 5.2e-12   8.9e-5    0.02   1388    1.0
    prob_n[186]    2.3e-3  1.6e-4  5.8e-3 6.9e-164  2.1e-28 4.5e-12   9.2e-5    0.02   1386    1.0
    prob_n[187]    2.3e-3  1.6e-4  5.8e-3 2.6e-165  8.4e-29 5.1e-12   9.5e-5    0.02   1384    1.0
    prob_n[188]    2.3e-3  1.6e-4  5.8e-3 9.6e-167  3.4e-29 5.6e-12   1.0e-4    0.02   1384    1.0
    prob_n[189]    2.3e-3  1.6e-4  5.8e-3 3.6e-168  1.3e-29 6.2e-12   1.1e-4    0.02   1385    1.0
    prob_n[190]    2.3e-3  1.5e-4  5.7e-3 1.3e-169  5.4e-30 6.4e-12   1.1e-4    0.02   1387    1.0
    prob_n[191]    2.2e-3  1.5e-4  5.7e-3 4.9e-171  2.1e-30 5.6e-12   1.0e-4    0.02   1390    1.0
    prob_n[192]    2.2e-3  1.5e-4  5.7e-3 1.8e-172  8.5e-31 4.8e-12   9.5e-5    0.02   1394    1.0
    prob_n[193]    2.2e-3  1.5e-4  5.7e-3 6.6e-174  3.3e-31 4.6e-12   8.9e-5    0.02   1400    1.0
    prob_n[194]    2.2e-3  1.5e-4  5.6e-3 2.4e-175  1.3e-31 4.0e-12   8.2e-5    0.02   1406    1.0
    prob_n[195]    2.2e-3  1.5e-4  5.6e-3 8.8e-177  5.2e-32 4.0e-12   8.6e-5    0.02   1413    1.0
    prob_n[196]    2.2e-3  1.5e-4  5.6e-3 3.2e-178  2.0e-32 4.5e-12   8.0e-5    0.02   1420    1.0
    prob_n[197]    2.2e-3  1.5e-4  5.5e-3 1.2e-179  7.9e-33 5.2e-12   7.7e-5    0.02   1428    1.0
    prob_n[198]    2.2e-3  1.5e-4  5.5e-3 4.2e-181  3.1e-33 5.4e-12   6.9e-5    0.02   1437    1.0
    prob_n[199]    2.2e-3  1.4e-4  5.5e-3 1.5e-182  1.2e-33 5.4e-12   6.6e-5    0.02   1445    1.0
    prob_n[200]    2.2e-3  1.4e-4  5.4e-3 5.4e-184  4.6e-34 4.2e-12   6.6e-5    0.02   1454    1.0
    prob_n[201]    2.2e-3  1.4e-4  5.4e-3 1.9e-185  1.8e-34 3.7e-12   6.9e-5    0.02   1463    1.0
    prob_n[202]    2.1e-3  1.4e-4  5.4e-3 6.9e-187  6.8e-35 4.1e-12   6.6e-5    0.02   1471    1.0
    prob_n[203]    2.1e-3  1.4e-4  5.4e-3 2.5e-188  2.6e-35 5.0e-12   6.6e-5    0.02   1479    1.0
    prob_n[204]    2.1e-3  1.4e-4  5.3e-3 8.8e-190  9.9e-36 5.0e-12   6.7e-5    0.02   1486    1.0
    prob_n[205]    2.1e-3  1.4e-4  5.3e-3 3.1e-191  3.8e-36 5.5e-12   6.6e-5    0.02   1492    1.0
    prob_n[206]    2.1e-3  1.4e-4  5.3e-3 1.1e-192  1.4e-36 4.7e-12   6.1e-5    0.02   1498    1.0
    prob_n[207]    2.1e-3  1.3e-4  5.2e-3 3.9e-194  5.4e-37 4.1e-12   6.7e-5    0.02   1503    1.0
    prob_n[208]    2.1e-3  1.3e-4  5.2e-3 1.4e-195  2.0e-37 4.4e-12   6.8e-5    0.02   1506    1.0
    prob_n[209]    2.1e-3  1.3e-4  5.2e-3 4.7e-197  7.7e-38 4.7e-12   7.0e-5    0.02   1509    1.0
    prob_n[210]    2.0e-3  1.3e-4  5.1e-3 1.6e-198  2.9e-38 5.9e-12   6.4e-5    0.02   1510    1.0
    prob_n[211]    2.0e-3  1.3e-4  5.1e-3 5.7e-200  1.1e-38 5.6e-12   6.1e-5    0.02   1510    1.0
    prob_n[212]    2.0e-3  1.3e-4  5.1e-3 2.0e-201  4.0e-39 6.5e-12   6.6e-5    0.02   1508    1.0
    prob_n[213]    2.0e-3  1.3e-4  5.0e-3 6.9e-203  1.5e-39 6.6e-12   6.6e-5    0.02   1506    1.0
    prob_n[214]    2.0e-3  1.3e-4  5.0e-3 2.4e-204  5.6e-40 6.9e-12   7.4e-5    0.02   1502    1.0
    prob_n[215]    2.0e-3  1.3e-4  5.0e-3 8.2e-206  2.1e-40 6.6e-12   8.1e-5    0.02   1497    1.0
    prob_n[216]    2.0e-3  1.3e-4  4.9e-3 2.8e-207  7.6e-41 7.1e-12   7.2e-5    0.02   1491    1.0
    prob_n[217]    1.9e-3  1.3e-4  4.9e-3 9.6e-209  2.8e-41 8.1e-12   7.7e-5    0.02   1484    1.0
    prob_n[218]    1.9e-3  1.3e-4  4.9e-3 3.3e-210  1.0e-41 9.4e-12   7.0e-5    0.02   1476    1.0
    prob_n[219]    1.9e-3  1.3e-4  4.9e-3 1.1e-211  3.8e-42 9.5e-12   6.3e-5    0.02   1467    1.0
    prob_n[220]    1.9e-3  1.3e-4  4.8e-3 3.8e-213  1.4e-42 9.4e-12   6.2e-5    0.02   1458    1.0
    prob_n[221]    1.9e-3  1.3e-4  4.8e-3 1.3e-214  5.0e-43 7.8e-12   6.4e-5    0.02   1447    1.0
    prob_n[222]    1.9e-3  1.3e-4  4.8e-3 4.4e-216  1.8e-43 8.7e-12   6.6e-5    0.02   1437    1.0
    prob_n[223]    1.9e-3  1.3e-4  4.7e-3 1.5e-217  6.7e-44 9.0e-12   5.9e-5    0.02   1425    1.0
    prob_n[224]    1.9e-3  1.3e-4  4.7e-3 5.0e-219  2.4e-44 9.9e-12   6.1e-5    0.02   1412    1.0
    prob_n[225]    1.8e-3  1.3e-4  4.7e-3 1.7e-220  8.7e-45 9.4e-12   6.5e-5    0.02   1398    1.0
    prob_n[226]    1.8e-3  1.3e-4  4.7e-3 5.6e-222  3.1e-45 1.1e-11   6.2e-5    0.02   1384    1.0
    prob_n[227]    1.8e-3  1.2e-4  4.6e-3 1.9e-223  1.1e-45 1.0e-11   5.7e-5    0.02   1370    1.0
    prob_n[228]    1.8e-3  1.2e-4  4.6e-3 6.2e-225  4.0e-46 9.3e-12   5.8e-5    0.02   1356    1.0
    prob_n[229]    1.8e-3  1.2e-4  4.6e-3 2.1e-226  1.4e-46 1.0e-11   6.2e-5    0.02   1343    1.0
    prob_n[230]    1.8e-3  1.2e-4  4.5e-3 6.9e-228  5.2e-47 1.1e-11   5.8e-5    0.02   1330    1.0
    prob_n[231]    1.8e-3  1.2e-4  4.5e-3 2.3e-229  1.8e-47 1.1e-11   6.0e-5    0.02   1317    1.0
    prob_n[232]    1.8e-3  1.2e-4  4.5e-3 7.5e-231  6.5e-48 1.0e-11   6.5e-5    0.02   1305    1.0
    prob_n[233]    1.8e-3  1.2e-4  4.5e-3 2.5e-232  2.3e-48 9.8e-12   6.6e-5    0.02   1293    1.0
    prob_n[234]    1.7e-3  1.2e-4  4.4e-3 8.1e-234  8.2e-49 1.0e-11   6.9e-5    0.02   1282    1.0
    prob_n[235]    1.7e-3  1.2e-4  4.4e-3 2.7e-235  2.9e-49 1.0e-11   6.8e-5    0.02   1272    1.0
    prob_n[236]    1.7e-3  1.2e-4  4.4e-3 8.8e-237  1.0e-49 9.3e-12   6.9e-5    0.02   1263    1.0
    prob_n[237]    1.7e-3  1.2e-4  4.4e-3 2.9e-238  3.6e-50 7.6e-12   6.9e-5    0.02   1254    1.0
    prob_n[238]    1.7e-3  1.2e-4  4.4e-3 9.3e-240  1.3e-50 6.9e-12   7.0e-5    0.02   1246    1.0
    prob_n[239]    1.7e-3  1.2e-4  4.3e-3 3.0e-241  4.4e-51 5.4e-12   6.0e-5    0.02   1240    1.0
    prob_n[240]    1.7e-3  1.2e-4  4.3e-3 9.9e-243  1.5e-51 6.1e-12   5.5e-5    0.02   1234    1.0
    prob_n[241]    1.7e-3  1.2e-4  4.3e-3 3.2e-244  5.4e-52 5.8e-12   5.4e-5    0.02   1228    1.0
    prob_n[242]    1.7e-3  1.2e-4  4.3e-3 1.0e-245  1.9e-52 5.9e-12   5.2e-5    0.02   1224    1.0
    prob_n[243]    1.7e-3  1.2e-4  4.3e-3 3.3e-247  6.5e-53 5.1e-12   5.0e-5    0.02   1221    1.0
    prob_n[244]    1.6e-3  1.2e-4  4.2e-3 1.1e-248  2.3e-53 4.3e-12   5.4e-5    0.02   1218    1.0
    prob_n[245]    1.6e-3  1.2e-4  4.2e-3 3.5e-250  7.8e-54 3.6e-12   5.4e-5    0.02   1217    1.0
    prob_n[246]    1.6e-3  1.2e-4  4.2e-3 1.1e-251  2.7e-54 3.8e-12   5.4e-5    0.02   1216    1.0
    prob_n[247]    1.6e-3  1.2e-4  4.2e-3 3.6e-253  9.3e-55 3.5e-12   4.8e-5    0.02   1216    1.0
    prob_n[248]    1.6e-3  1.2e-4  4.2e-3 1.1e-254  3.2e-55 3.0e-12   5.0e-5    0.02   1217    1.0
    prob_n[249]    1.6e-3  1.2e-4  4.2e-3 3.6e-256  1.1e-55 3.0e-12   5.0e-5    0.02   1218    1.0
    prob_n[250]    1.6e-3  1.2e-4  4.1e-3 1.2e-257  3.8e-56 3.1e-12   5.0e-5    0.02   1220    1.0
    prob_n[251]    1.6e-3  1.2e-4  4.1e-3 3.7e-259  1.3e-56 2.7e-12   4.9e-5    0.02   1223    1.0
    prob_n[252]    1.6e-3  1.2e-4  4.1e-3 1.2e-260  4.4e-57 2.2e-12   5.3e-5    0.02   1226    1.0
    prob_n[253]    1.6e-3  1.2e-4  4.1e-3 3.7e-262  1.5e-57 1.5e-12   5.2e-5    0.02   1229    1.0
    prob_n[254]    1.6e-3  1.2e-4  4.1e-3 1.2e-263  5.1e-58 1.3e-12   5.7e-5    0.02   1234    1.0
    prob_n[255]    1.5e-3  1.2e-4  4.0e-3 3.7e-265  1.7e-58 8.7e-13   5.3e-5    0.02   1233    1.0
    prob_n[256]    1.5e-3  1.1e-4  4.0e-3 1.2e-266  5.8e-59 6.0e-13   4.9e-5    0.02   1241    1.0
    prob_n[257]    1.5e-3  1.1e-4  4.0e-3 3.6e-268  2.0e-59 5.1e-13   4.8e-5    0.02   1248    1.0
    prob_n[258]    1.5e-3  1.1e-4  4.0e-3 1.1e-269  6.7e-60 5.3e-13   4.7e-5    0.02   1251    1.0
    prob_n[259]    1.5e-3  1.1e-4  4.0e-3 3.6e-271  2.2e-60 4.2e-13   5.0e-5    0.02   1255    1.0
    prob_n[260]    1.5e-3  1.1e-4  4.0e-3 1.1e-272  7.5e-61 2.9e-13   5.0e-5    0.02   1259    1.0
    prob_n[261]    1.5e-3  1.1e-4  3.9e-3 3.5e-274  2.5e-61 2.0e-13   5.0e-5    0.02   1263    1.0
    prob_n[262]    1.5e-3  1.1e-4  3.9e-3 1.1e-275  8.5e-62 1.6e-13   4.5e-5    0.02   1267    1.0
    prob_n[263]    1.5e-3  1.1e-4  3.9e-3 3.4e-277  2.8e-62 1.1e-13   4.7e-5    0.02   1271    1.0
    prob_n[264]    1.5e-3  1.1e-4  3.9e-3 1.1e-278  9.5e-63 7.4e-14   4.6e-5    0.02   1275    1.0
    prob_n[265]    1.5e-3  1.1e-4  3.9e-3 3.3e-280  3.2e-63 5.1e-14   4.5e-5    0.02   1278    1.0
    prob_n[266]    1.5e-3  1.1e-4  3.8e-3 1.0e-281  1.1e-63 3.5e-14   4.6e-5    0.02   1281    1.0
    prob_n[267]    1.4e-3  1.1e-4  3.8e-3 3.1e-283  3.5e-64 4.5e-14   5.1e-5    0.01   1283    1.0
    prob_n[268]    1.4e-3  1.1e-4  3.8e-3 9.6e-285  1.2e-64 3.1e-14   5.2e-5    0.01   1285    1.0
    prob_n[269]    1.4e-3  1.1e-4  3.8e-3 3.0e-286  3.9e-65 2.2e-14   5.7e-5    0.01   1287    1.0
    prob_n[270]    1.4e-3  1.0e-4  3.8e-3 9.1e-288  1.3e-65 1.5e-14   5.2e-5    0.01   1288    1.0
    prob_n[271]    1.4e-3  1.0e-4  3.7e-3 2.8e-289  4.2e-66 1.0e-14   5.0e-5    0.01   1289    1.0
    prob_n[272]    1.4e-3  1.0e-4  3.7e-3 8.6e-291  1.4e-66 6.9e-15   4.6e-5    0.01   1289    1.0
    prob_n[273]    1.4e-3  1.0e-4  3.7e-3 2.6e-292  4.6e-67 4.7e-15   4.6e-5    0.01   1289    1.0
    prob_n[274]    1.4e-3  1.0e-4  3.7e-3 8.0e-294  1.5e-67 3.2e-15   4.7e-5    0.01   1288    1.0
    prob_n[275]    1.4e-3  1.0e-4  3.7e-3 2.4e-295  4.9e-68 2.2e-15   4.6e-5    0.01   1286    1.0
    prob_n[276]    1.4e-3  1.0e-4  3.6e-3 7.5e-297  1.6e-68 1.5e-15   5.1e-5    0.01   1284    1.0
    prob_n[277]    1.4e-3  1.0e-4  3.6e-3 2.3e-298  5.3e-69 1.0e-15   5.2e-5    0.01   1281    1.0
    prob_n[278]    1.4e-3  1.0e-4  3.6e-3 6.9e-300  1.7e-69 6.8e-16   4.8e-5    0.01   1278    1.0
    prob_n[279]    1.4e-3  1.0e-4  3.6e-3 2.1e-301  5.6e-70 4.6e-16   4.7e-5    0.01   1275    1.0
    prob_n[280]    1.3e-3  1.0e-4  3.6e-3 6.3e-303  1.8e-70 3.1e-16   5.1e-5    0.01   1271    1.0
    prob_n[281]    1.3e-3  1.0e-4  3.6e-3 1.9e-304  6.0e-71 2.1e-16   5.7e-5    0.01   1266    1.0
    prob_n[282]    1.3e-3  1.0e-4  3.6e-3 5.8e-306  1.9e-71 1.4e-16   6.0e-5    0.01   1261    1.0
    prob_n[283]    1.3e-3 10.0e-5  3.5e-3 1.7e-307  6.3e-72 9.4e-17   5.9e-5    0.01   1256    1.0
    prob_n[284]    1.3e-3 10.0e-5  3.5e-3 5.2e-309  2.0e-72 6.3e-17   5.6e-5    0.01   1250    1.0
    prob_n[285]    1.3e-3  9.9e-5  3.5e-3      0.0  6.6e-73 4.2e-17   5.7e-5    0.01   1244    1.0
    prob_n[286]    1.3e-3  9.9e-5  3.5e-3      0.0  2.1e-73 2.8e-17   5.1e-5    0.01   1238    1.0
    prob_n[287]    1.3e-3  9.9e-5  3.5e-3      0.0  6.8e-74 1.9e-17   4.6e-5    0.01   1231    1.0
    prob_n[288]    1.3e-3  9.9e-5  3.5e-3      0.0  2.2e-74 1.3e-17   4.5e-5    0.01   1225    1.0
    prob_n[289]    1.3e-3  9.9e-5  3.4e-3      0.0  7.1e-75 8.3e-18   4.7e-5    0.01   1217    1.0
    prob_n[290]    1.3e-3  9.9e-5  3.4e-3      0.0  2.3e-75 5.5e-18   4.6e-5    0.01   1209    1.0
    prob_n[291]    1.3e-3  9.9e-5  3.4e-3      0.0  7.3e-76 3.7e-18   4.8e-5    0.01   1200    1.0
    prob_n[292]    1.3e-3  9.9e-5  3.4e-3      0.0  2.3e-76 2.4e-18   5.1e-5    0.01   1192    1.0
    prob_n[293]    1.3e-3  9.9e-5  3.4e-3      0.0  7.4e-77 1.6e-18   5.0e-5    0.01   1184    1.0
    prob_n[294]    1.3e-3  9.8e-5  3.4e-3      0.0  2.4e-77 1.1e-18   4.5e-5    0.01   1176    1.0
    prob_n[295]    1.3e-3  9.8e-5  3.4e-3      0.0  7.6e-78 7.0e-19   4.4e-5    0.01   1167    1.0
    prob_n[296]    1.3e-3  9.8e-5  3.3e-3      0.0  2.4e-78 4.6e-19   4.2e-5    0.01   1159    1.0
    prob_n[297]    1.3e-3  9.8e-5  3.3e-3      0.0  7.7e-79 3.1e-19   4.2e-5    0.01   1152    1.0
    prob_n[298]    1.2e-3  9.8e-5  3.3e-3      0.0  2.4e-79 2.0e-19   4.7e-5    0.01   1144    1.0
    prob_n[299]    1.2e-3  9.8e-5  3.3e-3      0.0  7.7e-80 1.3e-19   5.1e-5    0.01   1136    1.0
    prob_n[300]    1.2e-3  9.8e-5  3.3e-3      0.0  2.4e-80 8.6e-20   5.2e-5    0.01   1129    1.0
    prob_n[301]    1.2e-3  9.8e-5  3.3e-3      0.0  7.7e-81 5.7e-20   5.1e-5    0.01   1121    1.0
    prob_n[302]    1.2e-3  9.8e-5  3.3e-3      0.0  2.4e-81 3.7e-20   5.5e-5    0.01   1114    1.0
    prob_n[303]    1.2e-3  9.8e-5  3.3e-3      0.0  7.7e-82 2.4e-20   5.2e-5    0.01   1106    1.0
    prob_n[304]    1.2e-3  9.8e-5  3.2e-3      0.0  2.4e-82 1.6e-20   5.7e-5    0.01   1097    1.0
    prob_n[305]    1.2e-3  9.8e-5  3.2e-3      0.0  7.7e-83 1.0e-20   5.6e-5    0.01   1088    1.0
    prob_n[306]    1.2e-3  9.8e-5  3.2e-3      0.0  2.4e-83 6.7e-21   5.4e-5    0.01   1079    1.0
    prob_n[307]    1.2e-3  9.8e-5  3.2e-3      0.0  7.6e-84 4.4e-21   5.4e-5    0.01   1070    1.0
    prob_n[308]    1.2e-3  9.8e-5  3.2e-3      0.0  2.4e-84 2.8e-21   5.7e-5    0.01   1060    1.0
    prob_n[309]    1.2e-3  9.8e-5  3.2e-3      0.0  7.4e-85 1.8e-21   6.1e-5    0.01   1051    1.0
    prob_n[310]    1.2e-3  9.8e-5  3.2e-3      0.0  2.3e-85 1.2e-21   6.6e-5    0.01   1041    1.0
    prob_n[311]    1.2e-3  9.8e-5  3.2e-3      0.0  7.2e-86 7.7e-22   6.1e-5    0.01   1031    1.0
    prob_n[312]    1.2e-3  9.8e-5  3.1e-3      0.0  2.3e-86 5.0e-22   5.8e-5    0.01   1022    1.0
    prob_n[313]    1.2e-3  9.9e-5  3.1e-3      0.0  7.0e-87 3.2e-22   5.8e-5    0.01   1012    1.0
    prob_n[314]    1.2e-3  9.9e-5  3.1e-3      0.0  2.2e-87 2.1e-22   5.7e-5    0.01   1002    1.0
    prob_n[315]    1.2e-3  9.9e-5  3.1e-3      0.0  6.8e-88 1.3e-22   5.9e-5    0.01    992    1.0
    prob_n[316]    1.2e-3  9.9e-5  3.1e-3      0.0  2.1e-88 8.5e-23   5.9e-5    0.01    982    1.0
    prob_n[317]    1.2e-3  9.9e-5  3.1e-3      0.0  6.6e-89 5.5e-23   6.2e-5    0.01    972    1.0
    prob_n[318]    1.2e-3 10.0e-5  3.1e-3      0.0  2.0e-89 3.5e-23   6.4e-5    0.01    962    1.0
    prob_n[319]    1.2e-3  1.0e-4  3.1e-3      0.0  6.3e-90 2.3e-23   6.9e-5    0.01    952    1.0
    prob_n[320]    1.2e-3  1.0e-4  3.1e-3      0.0  1.9e-90 1.4e-23   7.5e-5    0.01    943    1.0
    prob_n[321]    1.2e-3  1.0e-4  3.1e-3      0.0  6.0e-91 9.2e-24   7.4e-5    0.01    933    1.0
    prob_n[322]    1.2e-3  1.0e-4  3.1e-3      0.0  1.9e-91 5.9e-24   7.5e-5    0.01    924    1.0
    prob_n[323]    1.2e-3  1.0e-4  3.1e-3      0.0  5.7e-92 3.8e-24   7.8e-5    0.01    913    1.0
    prob_n[324]    1.2e-3  1.0e-4  3.1e-3      0.0  1.8e-92 2.4e-24   7.9e-5    0.01    902    1.0
    prob_n[325]    1.2e-3  1.0e-4  3.1e-3      0.0  5.4e-93 1.5e-24   7.0e-5    0.01    890   1.01
    prob_n[326]    1.2e-3  1.0e-4  3.0e-3      0.0  1.7e-93 9.7e-25   7.2e-5    0.01    880   1.01
    prob_n[327]    1.2e-3  1.0e-4  3.0e-3      0.0  5.1e-94 6.2e-25   7.1e-5    0.01    869   1.01
    prob_n[328]    1.2e-3  1.0e-4  3.0e-3      0.0  1.6e-94 3.9e-25   7.3e-5    0.01    858   1.01
    prob_n[329]    1.2e-3  1.0e-4  3.0e-3      0.0  4.8e-95 2.5e-25   7.2e-5    0.01    848   1.01
    prob_n[330]    1.2e-3  1.0e-4  3.0e-3      0.0  1.5e-95 1.6e-25   7.1e-5    0.01    839   1.01
    prob_n[331]    1.2e-3  1.1e-4  3.0e-3      0.0  4.5e-96 9.9e-26   6.7e-5    0.01    826   1.01
    prob_n[332]    1.2e-3  1.1e-4  3.0e-3      0.0  1.4e-96 6.3e-26   6.8e-5    0.01    813   1.01
    prob_n[333]    1.2e-3  1.1e-4  3.0e-3      0.0  4.1e-97 4.0e-26   7.0e-5    0.01    802   1.01
    prob_n[334]    1.2e-3  1.1e-4  3.0e-3      0.0  1.3e-97 2.5e-26   7.3e-5    0.01    790   1.01
    prob_n[335]    1.2e-3  1.1e-4  3.0e-3      0.0  3.8e-98 1.6e-26   7.1e-5    0.01    780   1.01
    prob_n[336]    1.2e-3  1.1e-4  3.0e-3      0.0  1.2e-98 9.9e-27   6.8e-5    0.01    770   1.01
    prob_n[337]    1.2e-3  1.1e-4  3.0e-3      0.0  3.5e-99 6.2e-27   6.7e-5    0.01    760   1.01
    prob_n[338]    1.2e-3  1.1e-4  3.0e-3      0.0  1.1e-99 3.9e-27   6.1e-5    0.01    751   1.01
    prob_n[339]    1.2e-3  1.1e-4  3.0e-3      0.0 3.3e-100 2.4e-27   5.8e-5    0.01    743   1.01
    prob_n[340]    1.2e-3  1.1e-4  3.0e-3      0.0 9.8e-101 1.5e-27   5.8e-5    0.01    735   1.01
    prob_n[341]    1.2e-3  1.1e-4  3.0e-3      0.0 3.0e-101 9.6e-28   5.8e-5    0.01    727   1.01
    prob_n[342]    1.2e-3  1.1e-4  3.0e-3      0.0 9.0e-102 6.0e-28   5.4e-5    0.01    720   1.01
    prob_n[343]    1.2e-3  1.1e-4  3.0e-3      0.0 2.7e-102 3.7e-28   5.6e-5    0.01    714   1.01
    prob_n[344]    1.2e-3  1.1e-4  3.0e-3      0.0 8.2e-103 2.3e-28   5.1e-5    0.01    708   1.01
    prob_n[345]    1.2e-3  1.1e-4  3.0e-3      0.0 2.5e-103 1.5e-28   4.6e-5    0.01    702   1.01
    prob_n[346]    1.2e-3  1.1e-4  3.0e-3      0.0 7.4e-104 9.1e-29   4.2e-5    0.01    697   1.01
    prob_n[347]    1.2e-3  1.1e-4  3.0e-3      0.0 2.2e-104 5.7e-29   4.1e-5    0.01    692   1.01
    prob_n[348]    1.2e-3  1.1e-4  3.0e-3      0.0 6.7e-105 3.5e-29   3.8e-5    0.01    687   1.01
    prob_n[349]    1.2e-3  1.1e-4  3.0e-3      0.0 2.0e-105 2.2e-29   3.7e-5    0.01    683   1.01
    prob_n[350]    1.2e-3  1.2e-4  3.0e-3      0.0 6.0e-106 1.4e-29   3.8e-5    0.01    679   1.01
    prob_n[351]    1.2e-3  1.2e-4  3.0e-3      0.0 1.8e-106 8.4e-30   3.4e-5    0.01    676   1.01
    prob_n[352]    1.2e-3  1.2e-4  3.0e-3      0.0 5.4e-107 5.2e-30   3.3e-5    0.01    673   1.01
    prob_n[353]    1.2e-3  1.2e-4  3.0e-3      0.0 1.6e-107 3.2e-30   3.3e-5    0.01    669   1.01
    prob_n[354]    1.2e-3  1.2e-4  3.0e-3      0.0 4.8e-108 2.0e-30   3.6e-5    0.01    666   1.01
    prob_n[355]    1.2e-3  1.2e-4  3.0e-3      0.0 1.4e-108 1.2e-30   3.9e-5    0.01    664   1.01
    prob_n[356]    1.2e-3  1.2e-4  3.0e-3      0.0 4.3e-109 7.6e-31   4.0e-5    0.01    661   1.01
    prob_n[357]    1.2e-3  1.2e-4  3.0e-3      0.0 1.3e-109 4.7e-31   3.8e-5    0.01    658   1.01
    prob_n[358]    1.2e-3  1.2e-4  3.0e-3      0.0 3.8e-110 2.9e-31   3.7e-5    0.01    656   1.01
    prob_n[359]    1.2e-3  1.2e-4  3.0e-3      0.0 1.1e-110 1.8e-31   3.4e-5    0.01    654   1.01
    prob_n[360]    1.2e-3  1.2e-4  3.0e-3      0.0 3.3e-111 1.1e-31   3.0e-5    0.01    652   1.01
    prob_n[361]    1.2e-3  1.2e-4  3.0e-3      0.0 9.9e-112 6.7e-32   2.9e-5    0.01    650   1.01
    prob_n[362]    1.2e-3  1.2e-4  3.0e-3      0.0 2.9e-112 4.1e-32   3.2e-5    0.01    649   1.01
    prob_n[363]    1.2e-3  1.2e-4  3.0e-3      0.0 8.7e-113 2.5e-32   2.9e-5    0.01    648   1.01
    prob_n[364]    1.2e-3  1.2e-4  3.0e-3      0.0 2.6e-113 1.5e-32   2.6e-5    0.01    647   1.01
    prob_n[365]    1.2e-3  1.2e-4  2.9e-3      0.0 7.6e-114 9.5e-33   2.3e-5    0.01    646   1.01
    prob_n[366]    1.2e-3  1.2e-4  2.9e-3      0.0 2.2e-114 5.8e-33   2.0e-5    0.01    646   1.01
    prob_n[367]    1.2e-3  1.2e-4  2.9e-3      0.0 6.6e-115 3.5e-33   2.0e-5    0.01    646   1.01
    prob_n[368]    1.2e-3  1.2e-4  2.9e-3      0.0 2.0e-115 2.2e-33   1.9e-5    0.01    645   1.01
    prob_n[369]    1.2e-3  1.2e-4  2.9e-3      0.0 5.8e-116 1.3e-33   1.7e-5    0.01    644   1.01
    prob_n[370]    1.2e-3  1.2e-4  2.9e-3      0.0 1.7e-116 8.0e-34   1.5e-5    0.01    642   1.01
    prob_n[371]    1.2e-3  1.2e-4  2.9e-3      0.0 5.0e-117 4.9e-34   1.3e-5    0.01    640    1.0
    prob_n[372]    1.2e-3  1.2e-4  2.9e-3      0.0 1.5e-117 3.0e-34   1.3e-5    0.01    638    1.0
    prob_n[373]    1.2e-3  1.2e-4  2.9e-3      0.0 4.3e-118 1.8e-34   1.1e-5    0.01    636    1.0
    prob_n[374]    1.2e-3  1.2e-4  2.9e-3      0.0 1.3e-118 1.1e-34  10.0e-6    0.01    633    1.0
    prob_n[375]    1.2e-3  1.2e-4  2.9e-3      0.0 3.7e-119 6.7e-35   8.8e-6    0.01    631    1.0
    prob_n[376]    1.2e-3  1.2e-4  2.9e-3      0.0 1.1e-119 4.1e-35   7.8e-6    0.01    629    1.0
    prob_n[377]    1.2e-3  1.2e-4  2.9e-3      0.0 3.2e-120 2.5e-35   6.8e-6    0.01    627    1.0
    prob_n[378]    1.2e-3  1.2e-4  2.9e-3      0.0 9.3e-121 1.5e-35   6.5e-6    0.01    624    1.0
    prob_n[379]    1.2e-3  1.2e-4  2.9e-3      0.0 2.7e-121 9.0e-36   7.2e-6    0.01    622    1.0
    prob_n[380]    1.2e-3  1.2e-4  2.9e-3      0.0 7.9e-122 5.4e-36   7.9e-6    0.01    620    1.0
    prob_n[381]    1.2e-3  1.2e-4  2.9e-3      0.0 2.3e-122 3.3e-36   7.1e-6    0.01    618    1.0
    prob_n[382]    1.2e-3  1.2e-4  2.9e-3      0.0 6.8e-123 2.0e-36   6.3e-6    0.01    616    1.0
    prob_n[383]    1.2e-3  1.2e-4  2.9e-3      0.0 2.0e-123 1.2e-36   5.6e-6    0.01    614    1.0
    prob_n[384]    1.2e-3  1.2e-4  2.8e-3      0.0 5.7e-124 7.2e-37   4.9e-6    0.01    612    1.0
    prob_n[385]    1.2e-3  1.2e-4  2.8e-3      0.0 1.7e-124 4.4e-37   4.3e-6    0.01    610    1.0
    prob_n[386]    1.2e-3  1.2e-4  2.8e-3      0.0 4.8e-125 2.6e-37   3.8e-6    0.01    607    1.0
    prob_n[387]    1.2e-3  1.2e-4  2.8e-3      0.0 1.4e-125 1.6e-37   3.3e-6    0.01    605    1.0
    prob_n[388]    1.2e-3  1.2e-4  2.8e-3      0.0 4.1e-126 9.5e-38   2.9e-6    0.01    603    1.0
    prob_n[389]    1.2e-3  1.2e-4  2.8e-3      0.0 1.2e-126 5.7e-38   2.6e-6    0.01    602    1.0
    prob_n[390]    1.2e-3  1.2e-4  2.8e-3      0.0 3.4e-127 3.4e-38   2.3e-6    0.01    600    1.0
    prob_n[391]    1.2e-3  1.2e-4  2.8e-3      0.0 9.9e-128 2.0e-38   2.0e-6    0.01    598    1.0
    prob_n[392]    1.2e-3  1.2e-4  2.8e-3      0.0 2.9e-128 1.2e-38   1.7e-6    0.01    596    1.0
    prob_n[393]    1.2e-3  1.2e-4  2.8e-3      0.0 8.3e-129 7.3e-39   1.5e-6    0.01    594    1.0
    prob_n[394]    1.2e-3  1.2e-4  2.8e-3      0.0 2.4e-129 4.4e-39   1.3e-6    0.01    592    1.0
    prob_n[395]    1.2e-3  1.2e-4  2.8e-3      0.0 6.9e-130 2.6e-39   1.2e-6    0.01    591    1.0
    prob_n[396]    1.1e-3  1.2e-4  2.8e-3      0.0 2.0e-130 1.6e-39   1.0e-6    0.01    589    1.0
    prob_n[397]    1.1e-3  1.2e-4  2.8e-3      0.0 5.8e-131 9.3e-40   8.8e-7    0.01    587    1.0
    prob_n[398]    1.1e-3  1.2e-4  2.8e-3      0.0 1.7e-131 5.6e-40   7.7e-7    0.01    586    1.0
    prob_n[399]    1.1e-3  1.2e-4  2.8e-3      0.0 4.8e-132 3.3e-40   6.7e-7    0.01    584    1.0
    prob_n[400]    1.1e-3  1.2e-4  2.8e-3      0.0 1.4e-132 2.0e-40   5.8e-7    0.01    583    1.0
    prob_n[401]    1.1e-3  1.2e-4  2.8e-3      0.0 4.0e-133 1.2e-40   5.1e-7    0.01    581    1.0
    prob_n[402]    1.1e-3  1.2e-4  2.8e-3      0.0 1.1e-133 7.0e-41   4.4e-7    0.01    580    1.0
    prob_n[403]    1.1e-3  1.2e-4  2.8e-3      0.0 3.3e-134 4.2e-41   3.8e-7    0.01    579    1.0
    prob_n[404]    1.1e-3  1.2e-4  2.8e-3      0.0 9.3e-135 2.5e-41   3.3e-7    0.01    577    1.0
    prob_n[405]    1.1e-3  1.2e-4  2.8e-3      0.0 2.7e-135 1.5e-41   2.9e-7    0.01    576    1.0
    prob_n[406]    1.1e-3  1.2e-4  2.8e-3      0.0 7.7e-136 8.7e-42   2.5e-7    0.01    575    1.0
    prob_n[407]    1.1e-3  1.2e-4  2.8e-3      0.0 2.2e-136 5.2e-42   2.2e-7    0.01    573    1.0
    prob_n[408]    1.1e-3  1.2e-4  2.8e-3      0.0 6.3e-137 3.1e-42   1.9e-7    0.01    572    1.0
    prob_n[409]    1.1e-3  1.2e-4  2.8e-3      0.0 1.8e-137 1.8e-42   1.6e-7    0.01    571    1.0
    prob_n[410]    1.1e-3  1.2e-4  2.8e-3      0.0 5.1e-138 1.1e-42   1.4e-7    0.01    570    1.0
    prob_n[411]    1.1e-3  1.2e-4  2.7e-3      0.0 1.5e-138 6.3e-43   1.2e-7    0.01    569    1.0
    prob_n[412]    1.1e-3  1.2e-4  2.7e-3      0.0 4.2e-139 3.7e-43   1.0e-7    0.01    568    1.0
    prob_n[413]    1.1e-3  1.2e-4  2.7e-3      0.0 1.2e-139 2.2e-43   9.0e-8    0.01    567    1.0
    prob_n[414]    1.1e-3  1.2e-4  2.7e-3      0.0 3.4e-140 1.3e-43   7.8e-8    0.01    566    1.0
    prob_n[415]    1.1e-3  1.2e-4  2.7e-3      0.0 9.7e-141 7.7e-44   6.7e-8    0.01    565    1.0
    prob_n[416]    1.1e-3  1.2e-4  2.7e-3      0.0 2.7e-141 4.5e-44   5.8e-8    0.01    564    1.0
    prob_n[417]    1.1e-3  1.2e-4  2.7e-3      0.0 7.8e-142 2.7e-44   5.0e-8    0.01    563    1.0
    prob_n[418]    1.1e-3  1.2e-4  2.7e-3      0.0 2.2e-142 1.6e-44   4.3e-8    0.01    562    1.0
    prob_n[419]    1.1e-3  1.2e-4  2.7e-3      0.0 6.3e-143 9.2e-45   3.7e-8    0.01    561    1.0
    prob_n[420]    1.1e-3  1.2e-4  2.7e-3      0.0 1.8e-143 5.4e-45   3.2e-8    0.01    560    1.0
    prob_n[421]    1.1e-3  1.2e-4  2.7e-3      0.0 5.1e-144 3.2e-45   2.7e-8    0.01    559    1.0
    prob_n[422]    1.1e-3  1.2e-4  2.7e-3      0.0 1.4e-144 1.9e-45   2.3e-8    0.01    558    1.0
    prob_n[423]    1.1e-3  1.1e-4  2.7e-3      0.0 4.1e-145 1.1e-45   2.0e-8    0.01    557    1.0
    prob_n[424]    1.1e-3  1.1e-4  2.7e-3      0.0 1.2e-145 6.4e-46   1.7e-8    0.01    556    1.0
    prob_n[425]    1.0e-3  1.1e-4  2.7e-3      0.0 3.3e-146 3.7e-46   1.5e-8    0.01    555    1.0
    prob_n[426]    1.0e-3  1.1e-4  2.7e-3      0.0 9.2e-147 2.2e-46   1.3e-8    0.01    554    1.0
    prob_n[427]    1.0e-3  1.1e-4  2.7e-3      0.0 2.6e-147 1.3e-46   1.1e-8    0.01    553    1.0
    prob_n[428]    1.0e-3  1.1e-4  2.7e-3      0.0 7.3e-148 7.5e-47   9.1e-9    0.01    552    1.0
    prob_n[429]    1.0e-3  1.1e-4  2.7e-3      0.0 2.1e-148 4.4e-47   7.8e-9    0.01    551    1.0
    prob_n[430]    1.0e-3  1.1e-4  2.7e-3      0.0 5.8e-149 2.6e-47   6.7e-9    0.01    550    1.0
    prob_n[431]    1.0e-3  1.1e-4  2.7e-3      0.0 1.6e-149 1.5e-47   5.7e-9    0.01    549    1.0
    prob_n[432]    1.0e-3  1.1e-4  2.7e-3      0.0 4.6e-150 8.7e-48   4.8e-9    0.01    548    1.0
    prob_n[433]    1.0e-3  1.1e-4  2.7e-3      0.0 1.3e-150 5.1e-48   4.1e-9    0.01    547    1.0
    prob_n[434]    1.0e-3  1.1e-4  2.7e-3      0.0 3.7e-151 3.0e-48   3.5e-9    0.01    546    1.0
    prob_n[435]   10.0e-4  1.1e-4  2.7e-3      0.0 1.0e-151 1.7e-48   3.0e-9    0.01    545    1.0
    prob_n[436]    9.9e-4  1.1e-4  2.7e-3      0.0 2.9e-152 1.0e-48   2.5e-9    0.01    544    1.0
    prob_n[437]    9.9e-4  1.1e-4  2.7e-3      0.0 8.1e-153 5.8e-49   2.2e-9    0.01    543    1.0
    prob_n[438]    9.8e-4  1.1e-4  2.7e-3      0.0 2.3e-153 3.4e-49   1.8e-9    0.01    542    1.0
    prob_n[439]    9.8e-4  1.1e-4  2.7e-3      0.0 6.4e-154 2.0e-49   1.6e-9    0.01    541    1.0
    prob_n[440]    9.7e-4  1.1e-4  2.7e-3      0.0 1.8e-154 1.1e-49   1.3e-9    0.01    540    1.0
    prob_n[441]    9.7e-4  1.1e-4  2.7e-3      0.0 5.0e-155 6.6e-50   1.1e-9    0.01    539    1.0
    prob_n[442]    9.6e-4  1.1e-4  2.7e-3      0.0 1.4e-155 3.8e-50  9.5e-10    0.01    538    1.0
    prob_n[443]    9.6e-4  1.1e-4  2.7e-3      0.0 3.9e-156 2.2e-50  8.0e-10    0.01    537    1.0
    prob_n[444]    9.5e-4  1.1e-4  2.7e-3      0.0 1.1e-156 1.3e-50  6.8e-10    0.01    536    1.0
    prob_n[445]    9.5e-4  1.1e-4  2.7e-3      0.0 3.1e-157 7.4e-51  5.8e-10    0.01    535    1.0
    prob_n[446]    9.4e-4  1.1e-4  2.7e-3      0.0 8.6e-158 4.3e-51  4.9e-10    0.01    534    1.0
    prob_n[447]    9.4e-4  1.1e-4  2.6e-3      0.0 2.4e-158 2.5e-51  4.1e-10    0.01    533    1.0
    prob_n[448]    9.3e-4  1.1e-4  2.6e-3      0.0 6.7e-159 1.4e-51  3.5e-10    0.01    532    1.0
    prob_n[449]    9.3e-4  1.1e-4  2.6e-3      0.0 1.9e-159 8.3e-52  2.9e-10    0.01    531    1.0
    prob_n[450]    9.2e-4  1.1e-4  2.6e-3      0.0 5.2e-160 4.8e-52  2.5e-10    0.01    530    1.0
    prob_n[451]    9.2e-4  1.1e-4  2.6e-3      0.0 1.5e-160 2.8e-52  2.1e-10    0.01    529    1.0
    prob_n[452]    9.1e-4  1.2e-4  2.6e-3      0.0 4.0e-161 1.6e-52  1.8e-10    0.01    528    1.0
    prob_n[453]    9.1e-4  1.2e-4  2.6e-3      0.0 1.1e-161 9.2e-53  1.5e-10    0.01    527    1.0
    prob_n[454]    9.0e-4  1.2e-4  2.6e-3      0.0 3.1e-162 5.3e-53  1.2e-10    0.01    525    1.0
    prob_n[455]    9.0e-4  1.2e-4  2.6e-3      0.0 8.7e-163 3.0e-53  1.0e-10    0.01    522    1.0
    prob_n[456]    8.9e-4  1.2e-4  2.6e-3      0.0 2.4e-163 1.7e-53  8.8e-11    0.01    518    1.0
    prob_n[457]    8.9e-4  1.2e-4  2.6e-3      0.0 6.7e-164 1.0e-53  7.4e-11    0.01    515    1.0
    prob_n[458]    8.8e-4  1.2e-4  2.6e-3      0.0 1.9e-164 5.8e-54  6.2e-11    0.01    512    1.0
    prob_n[459]    8.8e-4  1.2e-4  2.7e-3      0.0 5.2e-165 3.3e-54  5.2e-11    0.01    508    1.0
    prob_n[460]    8.7e-4  1.2e-4  2.7e-3      0.0 1.4e-165 1.9e-54  4.4e-11    0.01    505    1.0
    prob_n[461]    8.7e-4  1.2e-4  2.7e-3      0.0 4.0e-166 1.1e-54  3.7e-11    0.01    501    1.0
    prob_n[462]    8.6e-4  1.2e-4  2.7e-3      0.0 1.1e-166 6.2e-55  3.1e-11    0.01    497    1.0
    prob_n[463]    8.6e-4  1.2e-4  2.7e-3      0.0 3.0e-167 3.6e-55  2.6e-11    0.01    493    1.0
    prob_n[464]    8.5e-4  1.2e-4  2.7e-3      0.0 8.4e-168 2.1e-55  2.2e-11    0.01    488    1.0
    prob_n[465]    8.5e-4  1.2e-4  2.7e-3      0.0 2.3e-168 1.2e-55  1.8e-11    0.01    484    1.0
    prob_n[466]    8.4e-4  1.2e-4  2.7e-3      0.0 6.4e-169 6.7e-56  1.5e-11    0.01    479    1.0
    prob_n[467]    8.4e-4  1.2e-4  2.7e-3      0.0 1.8e-169 3.8e-56  1.3e-11    0.01    474    1.0
    prob_n[468]    8.3e-4  1.2e-4  2.7e-3      0.0 4.9e-170 2.2e-56  1.1e-11    0.01    469    1.0
    prob_n[469]    8.3e-4  1.3e-4  2.7e-3      0.0 1.3e-170 1.3e-56  8.8e-12    0.01    464    1.0
    prob_n[470]    8.3e-4  1.3e-4  2.7e-3      0.0 3.7e-171 7.2e-57  7.3e-12    0.01    458    1.0
    prob_n[471]    8.2e-4  1.3e-4  2.7e-3      0.0 1.0e-171 4.1e-57  6.1e-12    0.01    453    1.0
    prob_n[472]    8.2e-4  1.3e-4  2.8e-3      0.0 2.8e-172 2.3e-57  5.1e-12    0.01    447    1.0
    prob_n[473]    8.2e-4  1.3e-4  2.8e-3      0.0 7.7e-173 1.3e-57  4.3e-12    0.01    441    1.0
    prob_n[474]    8.1e-4  1.3e-4  2.8e-3      0.0 2.1e-173 7.6e-58  3.5e-12    0.01    436   1.01
    prob_n[475]    8.1e-4  1.4e-4  2.8e-3      0.0 5.9e-174 4.3e-58  2.9e-12    0.01    430   1.01
    prob_n[476]    8.0e-4  1.4e-4  2.8e-3      0.0 1.6e-174 2.4e-58  2.5e-12    0.01    424   1.01
    prob_n[477]    8.0e-4  1.4e-4  2.8e-3      0.0 4.4e-175 1.4e-58  2.0e-12    0.01    417   1.01
    prob_n[478]    8.0e-4  1.4e-4  2.9e-3      0.0 1.2e-175 7.9e-59  1.7e-12    0.01    411   1.01
    prob_n[479]    8.0e-4  1.4e-4  2.9e-3      0.0 3.3e-176 4.5e-59  1.4e-12    0.01    405   1.01
    prob_n[480]    7.9e-4  1.5e-4  2.9e-3      0.0 9.2e-177 2.6e-59  1.2e-12    0.01    399   1.01
    prob_n[481]    7.9e-4  1.5e-4  2.9e-3      0.0 2.5e-177 1.5e-59  9.7e-13    0.01    392   1.01
    prob_n[482]    7.9e-4  1.5e-4  3.0e-3      0.0 6.9e-178 8.2e-60  8.1e-13    0.01    386   1.01
    prob_n[483]    7.9e-4  1.5e-4  3.0e-3      0.0 1.9e-178 4.7e-60  6.7e-13    0.01    380   1.01
    prob_n[484]    7.8e-4  1.6e-4  3.1e-3      0.0 5.2e-179 2.6e-60  5.5e-13    0.01    373   1.01
    prob_n[485]    7.8e-4  1.6e-4  3.1e-3      0.0 1.4e-179 1.5e-60  4.6e-13    0.01    367   1.01
    prob_n[486]    7.8e-4  1.7e-4  3.1e-3      0.0 3.9e-180 8.5e-61  3.8e-13    0.01    361   1.01
    prob_n[487]    7.8e-4  1.7e-4  3.2e-3      0.0 1.1e-180 4.8e-61  3.1e-13    0.01    355   1.01
    prob_n[488]    7.8e-4  1.7e-4  3.2e-3      0.0 2.9e-181 2.7e-61  2.6e-13    0.01    349   1.01
    prob_n[489]    7.8e-4  1.8e-4  3.3e-3      0.0 7.9e-182 1.5e-61  2.1e-13    0.01    343   1.01
    prob_n[490]    7.8e-4  1.8e-4  3.3e-3      0.0 2.2e-182 8.7e-62  1.8e-13    0.01    338   1.01
    prob_n[491]    7.8e-4  1.9e-4  3.4e-3      0.0 5.9e-183 4.9e-62  1.5e-13    0.01    332   1.01
    prob_n[492]    7.8e-4  1.9e-4  3.5e-3      0.0 1.6e-183 2.8e-62  1.2e-13    0.01    327   1.01
    prob_n[493]    7.8e-4  2.0e-4  3.5e-3      0.0 4.4e-184 1.6e-62 10.0e-14    0.01    322   1.01
    prob_n[494]    7.8e-4  2.0e-4  3.6e-3      0.0 1.2e-184 8.8e-63  8.2e-14    0.01    317   1.01
    prob_n[495]    7.8e-4  2.1e-4  3.7e-3      0.0 3.2e-185 5.0e-63  6.8e-14    0.01    312   1.01
    prob_n[496]    7.8e-4  2.1e-4  3.8e-3      0.0 8.8e-186 2.8e-63  5.6e-14    0.01    307   1.01
    prob_n[497]    7.8e-4  2.2e-4  3.8e-3      0.0 2.4e-186 1.6e-63  4.6e-14    0.01    303   1.01
    prob_n[498]    7.8e-4  2.3e-4  3.9e-3      0.0 6.5e-187 8.9e-64  3.8e-14    0.01    299   1.01
    prob_n[499]    7.8e-4  2.3e-4  4.0e-3      0.0 1.8e-187 5.0e-64  3.1e-14    0.01    294   1.01
    prob_n[500]    7.8e-4  2.4e-4  4.1e-3      0.0 4.8e-188 2.8e-64  2.6e-14 10.0e-3    291   1.01
    lp__           -19.19    0.03    0.53   -20.55   -19.24  -19.01   -18.92  -18.87    369   1.01

    Samples were drawn using NUTS at Tue Feb 25 10:53:16 2020.
    For each parameter, n_eff is a crude measure of effective sample size,
    and Rhat is the potential scale reduction factor on split chains (at
    convergence, Rhat=1).

```python

plt.hist(fit['theta'])
plt.xlabel('Theta')
plt.ylabel('Number of Samples')
plt.show

```

```python

plt.plot(fit['theta'], fit['n'], '.')
plt.xlabel('Theta')
plt.ylabel('Number of Samples')
plt.show
```

 Finally let us consider a coin tossing example from earlier, but in this case we will have run the experiment twice. We will then use data from both experiments to fit our head rates.

```python
two_exp_rate = """// Inferring a Common Rate, With Posterior Predictive
data {
  int<lower=1> n1;
  int<lower=1> n2;
  int<lower=0> k1;
  int<lower=0> k2;
}
parameters {
  real<lower=0,upper=1> theta;
}
model {
  // Prior on Single Rate Theta
  theta ~ beta(1, 1);

  // Observed Counts
  k1 ~ binomial(n1, theta);
  k2 ~ binomial(n2, theta);
}"""

sm=pystan.StanModel(model_code=two_exp_rate)
```

    INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_bd8f6a5f874a46897ced59e716bf395f NOW.

```python
data_1 = {
'k1': 4,
'k2': 6,
'n1': 10,
'n2': 10}

data_2 = {
'k1': 0,
'k2': 10,
'n1': 10,
'n2': 10}
```

```python
fit1=sm.sampling(data=data_1,chains=4,iter=10000,seed=12329)

plt.hist(fit1['theta'])
plt.show()

fit2=sm.sampling(data=data_2,chains=4,iter=10000,seed=12329)

plt.hist(fit2['theta'])
plt.show()

```

```python
two_exp_rate_gen = """// Inferring a Common Rate, With Posterior Predictive
data {
  int<lower=1> n1;
  int<lower=1> n2;
  int<lower=0> k1;
  int<lower=0> k2;
}
parameters {
  real<lower=0,upper=1> theta;
}
model {
  // Prior on Single Rate Theta
  theta ~ beta(1, 1);

  // Observed Counts
  k1 ~ binomial(n1, theta);
  k2 ~ binomial(n2, theta);
}
generated quantities {
  int<lower=0,upper=n1> postpredk1;
  int<lower=0,upper=n2> postpredk2;

  // Posterior Predictive
  postpredk1 <- binomial_rng(n1, theta);
  postpredk2 <- binomial_rng(n2, theta);
}"""

sm=pystan.StanModel(model_code=two_exp_rate_gen)
```

    INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_1070d695aa6003acbae0397bae8b9e8b NOW.

```python
fit1=sm.sampling(data=data_1,chains=4,iter=10000,seed=12329)

plt.hist(fit1['theta'])
plt.show()

fit2=sm.sampling(data=data_2,chains=4,iter=10000,seed=12329)

plt.hist(fit2['theta'])
plt.show()

```

```python
plt.hist(fit1['postpredk1'])
```

```python
heatmap = np.zeros([11,11])

for x in np.arange(2000):
    heatmap[int(fit2['postpredk1'][x]), int(fit2['postpredk2'][x])] += 1
```

```python
plt.imshow(heatmap, cmap='hot', interpolation='nearest')
plt.show()
```

 Not only is it useful to understand your model before you fit it, but to understand how well your data is explained by your model. Part of this process is creative but there are also tools that exist that can help.

```python

```

```python
x=np.random.normal(3,1,10)
```

```python
x
```

    array([4.48581462, 3.935212  , 2.13096659, 2.43710064, 5.40903363,
           2.33496308, 1.64093622, 2.86598126, 3.23251945, 2.31631115])

```python
model="""
data {
int<lower=0> N; //number of datapoints
real x[N]; //x values
}

parameters {
real mu; //mean of normal distribution
real<lower=0.0> sigma;
}

model {

sigma ~ cauchy(0,1);
x ~ normal(mu,sigma);
}
"""
```

```python
sm=pystan.StanModel(model_code=model)
```

    INFO:pystan:COMPILING THE C++ CODE FOR MODEL anon_model_23840a14d34f49990979ae476b61c011 NOW.

```python
data={
    'N':10,
    'x':x
}
```

```python
fit=sm.sampling(data=data,chains=4)
```

    WARNING:pystan:1 of 4000 iterations saturated the maximum tree depth of 10 (0.025 %)
    WARNING:pystan:Run again with max_treedepth larger than 10 to avoid saturation

```python
fit
```

    Inference for Stan model: anon_model_23840a14d34f49990979ae476b61c011.
    4 chains, each with iter=2000; warmup=1000; thin=1;
    post-warmup draws per chain=1000, total post-warmup draws=4000.

            mean se_mean     sd   2.5%    25%    50%    75%  97.5%  n_eff   Rhat
    mu      3.07  9.2e-3   0.42   2.24   2.79   3.07   3.34   3.93   2136    1.0
    sigma   1.29  7.5e-3   0.34   0.82   1.06   1.22   1.45   2.13   1993    1.0
    lp__   -8.02    0.03    1.1 -11.03  -8.45   -7.7  -7.23  -6.93   1308    1.0

    Samples were drawn using NUTS at Tue Feb 25 12:30:50 2020.
    For each parameter, n_eff is a crude measure of effective sample size,
    and Rhat is the potential scale reduction factor on split chains (at
    convergence, Rhat=1).

```python
plt.hist(fit['sigma'])
```

```python
plt.scatter(fit['mu'],fit['sigma'])
```

```python
import seaborn as sns
import pandas as pd
```

```python
df=pd.DataFrame(np.vstack((fit['mu'],fit['sigma'])).T,columns=[r'$\mu$',r'$\sigma$'])

```

```python
df[0:5]
```

      $\mu$
      $\sigma$

      0
      2.721063
      1.731366

      1
      2.950122
      1.320691

      2
      3.210656
      1.188615

      3
      3.230699
      1.271445

      4
      3.594663
      1.501862

```python
g=sns.PairGrid(df)
g.map_diag(plt.hist)
g.map_lower(sns.kdeplot)
```

```python

```