Illustration with Python: Chebyshev’s Inequality

Chebyshev’sInequality.py

import numpy as np
import random
import matplotlib.pyplot as plt

# Step 1
# create a population with a gamma distribution
shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
mu = shape*scale # mean and standard deviation
sigma = scale*np.sqrt(shape)
s = np.random.gamma(shape, scale, 1000000)

# Step 2
# sample 10,000 sample
rs = random.choices(s, k=10000)

# Step 3
# set k
ks = [0.1,0.5,1.0,1.5,2.0,2.5,3.0]
# prob list
probs = []
# for each k
for k in ks: 
    # start count
    c = 0
    # for each data sample
    for i in rs:
        # count if far from mean in k standard deviation
        if abs(i - mu) > k * sigma :
            c += 1
    # probability of sample has a distance from an expected value larger than k standrad deviation
    probs.append(c/10000)

# Step 4
# set figure size
plot = plt.figure(figsize=(20,10))
# plot each probability
plt.plot(ks,probs, marker='o')
# show plot
plot.show()
# print each probability
print("Probability of a sample far from mean more than k standard deviation:")
for i, prob in enumerate(probs):
    print("k:" + str(ks[i]) + ", probability: " \
          + str(prob)[0:5] + \
          " | in theory, probability should less than: " \
          + str(1/ks[i]**2)[0:5])
input()