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 X_{1}, X_{2}, X_{3},……X_{n} are random variables drawn from any probability distribution function with mean Σμ_{i} and standard deviation Σσ_{i} where (i=1,2,3,……n). The sum of random variables X i.e X=X_{1}+ X_{2} + X_{3}+……+X_{n} with mean μ=Σμ_{i} and standard deviation σ=Σσ_{i} 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

## 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**

**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

*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.

**References**

**References**