2019-01-03[Stay Sharp]Gaussian Distribution and CLT

The Gaussian Distribution

The Gaussian Distribution (Normal distribution). For a single variable x, the Gaussian distribution's equation is

The corresponding of The Gaussian Distribution is

image.png

which is generated by the code

import matplotlib.pyplot as plt, matplotlib.mlab as mlab
import numpy as np
import math

mu = 0
variance = 1
sigma = math.sqrt(variance)
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
plt.plot(x, mlab.normpdf(x, mu, sigma))
plt.show()

The Central Limit Theorem

WikiPedia says, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends to a Gaussian Distribution, even if the original variables themselves are not normally distributed.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
from wand.image import Image
from wand.display import display

n = 1000

avg = []
for i in range(1, n):
    a = np.random.randint(1, 7, 10)
    avg.append(np.average(a))

def clt(current):
    plt.cla()
    if current == 1000:
        a.event_source.stop()

    plt.hist(avg[0:current])
    plt.gca().set_title('Expected value of die rolls')
    plt.gca().set_xlabel('Average from die roll')
    plt.gca().set_ylabel('Frequency')

    plt.annotate('Die roll = {}'.format(current), [3, 27])

fig = plt.figure()
a = animation.FuncAnimation(fig, clt, interval=1)

References

https://medium.freecodecamp.org/how-to-visualize-the-central-limit-theorem-in-python-b619f5b00168