Overview

Roughly introduce the relationship of each distribution
Python code creation to calculate and visualize the distribution
Execute under different conditions and confirm that the approximation error may be large or small.

Reference book

Basic Statistics

Roughly the relationship of each distribution

Hypergeometric and binomial distributions

In the hypergeometric distribution

What is the probability of having 3 red balls when 10 balls are taken out of a box containing 20 red and blue balls each? I think about the problem. The probability of red appearing the first time is 50%, but the probability changes the second time because the number of remaining red and blue changes due to the influence of the first time. This is called a non-restoring extract and each attempt is not independent (Reference: biostatistics --hypergeometric distribution)

On the other hand, in the binomial distribution

What is the probability that red will come out 3 times when you repeat the trial of taking out one ball from the box containing one red and blue ball and returning it 10 times? I think about the problem. Each trial is independent and is a matter of restore extraction, as each trial reverts to its original state. A common example is the same as "What is the probability of throwing a coin 10 times and getting a table 3 times?" (Reference: Statistics WEB-Binomial distribution)

The hypergeometric distribution can be well approximated by the binomial distribution if the total number (the number of balls in the box) is sufficiently larger than the number of extracts.

Binomial distribution, Poisson distribution and normal distribution

The binomial distribution is

If the number of trials (the number of coins tossed) is large enough, a normal distribution can be used for a good approximation.
If the probability of one trial is extremely small (for example, a coin that is distorted and appears only once in 1000 times), it can be approximated well by the Poisson distribution. (Approximate because it makes the calculation easier)

Calculation of distribution and confirmation of approximation error

Python code (function part)

import scipy as sc
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import japanize_matplotlib

# functions ------------------------
def binomial(n, x, p):
    '''
Binomial distribution
Event with probability p occurs x times in n trials
Each trial is independent
Example: An event where 3 is rolled on the dice (probability p)=1/6) but n dice=X when shaken 10 times=5 times)
        nCx * p^x * (1-p)^(n-x)
    '''
    return (
        sc.special.comb(n, x)
        * np.power(p, x)
        * np.power(1 - p, n - x)
    )

def poisson(n, x, p):
    '''
Poisson distribution
        p->When 0, the binomial distribution becomes a Poisson distribution.
    '''
    mean = n * p
    return (
        np.power(mean, x)
        * np.exp(-mean)
        / sc.special.factorial(x)
    )

def gaussian(n, x, p):
    '''
normal distribution
        n->When oo, the binomial distribution is normal
    '''
    mean = n * p
    variance = np.power(n * p * (1 - p), 1/2)
    return (
        1.0 
        / (np.power(2 * np.pi, 1/2) * variance)
        / np.exp(
            np.power(x - mean, 2)
            / (2 * np.power(variance, 2))
            )
    )

def hypergeometric(n, x, p, N):
    '''
Hypergeometric distribution
Each trial is not independent
It is necessary to consider all N for non-restoration extraction
    '''
    target_size = N * p  #Number of hits in the whole
    return (
        sc.special.comb(target_size, x)  #X combinations per
        * sc.special.comb(N - target_size, n - x)  #Lost n-x combinations
        / sc.special.comb(N, n)  #All combinations of n
    )


def calc_distributions(n, p, cols, N=0):
    '''
    ex.:
    n = 30  #Number of trials
    p = 0.16  #Percentage of hits
    N = 1000  #All (used only in hypergeometric distributions)
    '''
    #Prepare an array for storing results
    bi_arr = np.zeros(n + 1, dtype=np.float)
    po_arr = np.zeros(n + 1, dtype=np.float)
    ga_arr = np.zeros(n + 1, dtype=np.float)

    for x in np.arange(n + 1):
        #Store the result in an array
        bi_arr[x] = binomial(n, x, p) 
        po_arr[x] = poisson(n, x, p) 
        ga_arr[x] = gaussian(n, x, p) 

    #Store the result in a data frame
    df = pd.DataFrame()
    df['Binomial distribution'] = bi_arr
    df['Poisson distribution'] = po_arr
    df['normal distribution'] = ga_arr

    if N > 0:
        hy_arr = np.zeros(n + 1, dtype=np.float)
        for x in np.arange(n + 1):
            hy_arr[x] = hypergeometric(n, x, p, N)
        df['Hypergeometric distribution'] = hy_arr

    return df[cols]


def visualize(df,n, p, N=0):
    plot_params = {
        'linestyle':'-', 
        'markersize':5,
        'marker':'o'
    }
    colors = [
        'black',
        'blue',
        'green',
        'purple',
    ]
    
    plt.close(1)
    figure_ = plt.figure(1, figsize=(8,4))    
    axes_ = figure_.add_subplot(111)  #Create Axes
    for i, col in enumerate(df.columns):
        if i == 0:
            plot_params['linewidth'] = 2
            plot_params['alpha'] = 1.0
        else:
            plot_params['linewidth'] = 1
            plot_params['alpha'] = 0.4
        plot_params['color'] = colors[i]
        axes_.plot(
            df.index.values, df[col].values,
            label = col,
            **plot_params,
        )

    plt.legend()
    plt.xlabel('Number of hits x')
    plt.ylabel('Probability of winning x times out of n times')
    title = 'Percentage of hits p:{p:.2f},Number of trials n:{n}'.format(p=p, n=n)
    if 'Hypergeometric distribution' in df.columns:
        title += ',All N:{N}'.format(N=N)
    plt.title(title)
    xmax = n * p * 2
    if xmax<10: xmax = 10;
    plt.xlim([0, xmax])
    #(The expression "number of times" is not good considering non-restoration extraction)

Execute under different conditions, check error

Roll the dice 100 times and how many times 1 comes out

n = 100; p = 0.167
df = calc_distributions(n, p, cols=['Binomial distribution', 'Poisson distribution', 'normal distribution'])
visualize(df, n, p)

Binomial distribution and normal distribution match well

Shake the 100 facet 100 times and how many times 1 will appear

n = 100; p = 0.01
df = calc_distributions(n, p, cols=['Binomial distribution', 'Poisson distribution', 'normal distribution'])
visualize(df, n, p)

Binomial distribution and Poisson distribution match well

Of the 5,000 inhabitants, 10% are children. How many children are there out of 100 selected?

n = 100; p = 0.1; N = 5000
df = calc_distributions(n, p, cols=['Hypergeometric distribution','Binomial distribution'], N=N)
visualize(df, n, p, N)

Hypergeometric distribution and binomial distribution match well

10% of the 500 inhabitants are children. How many children are there out of 100 selected?

n = 100; p = 0.1; N = 500
df = calc_distributions(n, p, cols=['Hypergeometric distribution','Binomial distribution'], N=N)
visualize(df, n, p, N)

The error is noticeable compared to the case of N = 5000

end

Relationship and approximation error of binomial distribution, Poisson distribution, normal distribution, hypergeometric distribution