# Central Limit Theorem and Normal Distribution

Why is normal distribution is important?
To understand the question you have to go through the Central Limit Theorem.

## Central Limit Theorem

According to central limit theorem if X1, X2, X3,……Xn are random variables drawn from any probability distribution function with mean  Σμi  and standard deviation Σσi where (i=1,2,3,……n). The sum of random variables X i.e X=X1+ X2 + X3+……+Xn with mean μ=Σμi and standard deviation σ=Σσi will approach to normal distribution.

Due to this theorem, this continuous probability distribution function is very popular  and has several applications in variety of fields.

## Normal Distribution

A random variable X is said to follow normal distribution with two parameters μ and σ and is denoted by X~N(μ, σ²).  The normal distribution is also known as Gaussian distribution.

If it follows the following distribution function .

Further, a normal distribution   with normal variate Z is called standard normal distribution with mean μ=0 and standard deviation σ=1  i.e Z~N(0,1).
and

## Z= (X-μ)/ σ

Normal Distribution Formula

Normal Distribution Curve

## Properties of Normal Distribution

• Normal distribution curve is a bell shaped.
• Normal distribution curve is symmetrical  in which mean=median=mode and coincides at center.
• Skewness of normal distribution curve is 0.
• The total area under normal distribution curve is 1.

## Solved  Numerical Problems Related to Normal Distribution

Q-  If  X is normally distributed with mean 2 and standard deviation 9 then the calculate the probability distribution

P( 2<=X<=3) .

Solution-

In question

μ=2

and

σ²=1 i.e σ=1

Calculate Z using  formula Z= (X-μ)/ σ  for X=2

The we have Z=(2-2)/1= 0

Calculate Z using  formula Z= (X-μ)/ σ  for X=3

We get  Z=(3-2)/1=1

Then we get probability distribution  P(0<=Z<=1) corresponding to P( 2<=X<=3).

Here

P(0<=Z<=1) = P(Z<=1) – P(Z<=0)

The table Z (Click on the link to see Z table https://www.ztable.net/ ) to calculate are under  P(Z<=1) – P(Z<=0)

=(0.50+0.3413)-(0.5+0.0)= 0.3413

You can see area covered by  P(0<=Z<=1) is  0.3413

## Python Code to Plot Area Under Normal Distribution Curve

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
mu=0
sigma=1
x = np.arange(-4,4,0.001)
y = norm.pdf(x, mu, sigma)
z = x[(0 < x) & (x < 1.0)]
plt.plot(x, y)
plt.fill_between(z, 0, norm.pdf(z, mu, sigma))
plt.savefig(“Normal Distribution.png”)

The output of the program would be the image

Area Under Normal Distribution Curve

## Applications of Normal Distribution in Data Science and Machine Learning

• SVM (Support Vector Machine) uses Gaussian kernel which is  based on normal distribution.
• Gaussian Naive Bays classifier uses normal or Gaussian  distribution.
• For hypothetical testing in statistics.

## Conclusion-

In this post, I have explained about normal distribution or Gaussian distribution which is a very famous continuous probability distribution function. I has lot applications in machine learning and data science. Hope you will understand and apply it.