central_limit_and_large_numbers_01.py

import math
import matplotlib.pyplot as plt
import numpy as np
import random
import scipy.stats as stats

die_e = 3.5 # expected value of an ordinary six-sided die
die_var = 17.5/6 # variance of an ordinary six-sided die
die_sigma = math.sqrt(die_var)

def create_statistics():
    clt_sum_frequencies = np.zeros(601, dtype = int)
    lln_averages = []
    total_sum = 0
    for repetition in range(0, 10000):
        sum = 0
        for cast in range(0, 100):
            die_number = random.randint(1, 6)
            sum = sum + die_number
        clt_sum_frequencies[sum] += 1
        total_sum += sum
        lln_averages.append(total_sum / (100 * (repetition + 1)))
    return clt_sum_frequencies / 10000, lln_averages
  
def plot_clt_chart(sum_frequencies, shift, scale):
    sums = np.linspace(
        (0 - shift) / scale, 
        (600 - shift) / scale, 
        len(sum_frequencies))
    width = sums[1] - sums[0]
    sum_densities = sum_frequencies / width
    mu = 100 * die_e - shift
    sigma = die_sigma * math.sqrt(100) / scale
    normal_densities = [stats.norm.pdf(x, mu, sigma) for x in sums]
    min_x = 280
    max_x = 420
    plot_chart(
        sums[min_x:max_x],
        sum_densities[min_x:max_x],
        normal_densities[min_x:max_x], 
        width)
    
def plot_lln_chart(averages):
    mu = die_e
    indexes = np.arange(len(averages))
    plt.ylim(3.48, 3.52)
    plot_chart(indexes, averages, [mu for x in indexes])
    
def plot_chart(x_values, y_values, y_comparison_values, width=1.0):
    plt.bar(x_values, y_values, width=width, align='center')
    plt.plot(x_values, y_comparison_values, color='r')
    plt.show()
    
clt_sum_frequencies, lln_averages = create_statistics()
plot_clt_chart(clt_sum_frequencies, 0, 1)
plot_lln_chart(lln_averages)