Verify the Central Limit Theorem by Simulation

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Assignment for ECEN 646 Homework 5 Problem 4. Coded in Python.

Problem:

Let $ \{ X_i \}_{i>=1} $ be i.i.d. uniform (0, 1) random variables. Compute the standard deviation \(\sigma\) of \(X_i\) and define

\[ Z_N = \sum_{i=1}^N = \frac{X_i - E[X_i]}{\sigma\sqrt{N}} \]

Plot the empirical distribution functions (based on \(10^4\) random variables) for \(Z_{10}\), \(Z_{100}\) and \(Z_{1000}\). On the same figure plot the distribution function of the standard normal random variable.

Code:

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import matplotlib.pyplot as plt
import random
from random import seed
from statistics import *
from scipy.stats import norm

def z_n_calc(n):
    root_n = pow(n, 0.5)
    seed()
    num_array = [random.uniform(0,1) for _ in range(n)]
    expectation = sum(num_array)/n
    stdeviation = stdev(num_array)
    res = 0
    
    for i in range(0, n):
        res = res + (num_array[i]-expectation)/(stdeviation*root_n)

    return res

def sample_generator(n):
    return [z_n_calc(n) for _ in range(10000)]

def norm_cdf_generator(array_in):
    return [norm.cdf(x) for x in array_in]
    
    
y_array = [x/10000.0 for x in range(10000)]

# Z10
x_array_10 = sorted(sample_generator(10))
plt.figure()
# plt.axis([-2*(10**(-13)),2*(10**(-13)),0,1])
plt.plot(x_array_10, y_array, label='Z_10')
plt.plot(x_array_10, norm_cdf_generator(x_array_10))
plt.legend()

# Z100
x_array_100 = sorted(sample_generator(100))
plt.figure()
# plt.axis([-2*(10**(-13)),2*(10**(-13)),0,1])
plt.plot(x_array_100, y_array, label='Z_100')
plt.plot(x_array_100, norm_cdf_generator(x_array_100))
plt.legend()

# Z1000
x_array_1000 = sorted(sample_generator(1000))
plt.figure()
plt.plot(x_array_1000, y_array, label='Z_1000')
plt.plot(x_array_1000, norm_cdf_generator(x_array_1000))
plt.legend()

# All together
plt.figure()
plt.plot(x_array_10, y_array, label='Z_10')
plt.plot(x_array_100, y_array, label='Z_100')
plt.plot(x_array_1000, y_array, label='Z_1000')
plt.plot(x_array_1000, norm_cdf_generator(x_array_10), label='Standard normal')
plt.legend()

plt.ylabel('CDF')
plt.show()

Plot:

Z_10 plot

\(Z_{10}\) plot

Z_100 plot

\(Z_{100}\) plot

Z_1000 plot

\(Z_{1000}\) plot

All together plot

All plot

GitHub repository