# Probability Density Function Tutorial

## Probability Density Function Tutorial

##### Definition:

The Probability Density Function(PDF) of a continuous random variable is a function which can be integrated to obtain the probability that the random variable takes a value in a given interval. PDF is used to find the point of Normal Distribution curve. Continuous Probability Density Function of the Normal Distribution is called the Gaussian Function.

##### Standard Normal Distribution:

A random variable which has a normal distribution with a mean m=0 and a standard deviation σ=1 is referred to as Standard Normal Distribution.

#### Formula:

PDF of Normal Distribution = P(x) = (1/(σsqrt(2π)))e-(x-m)2 / (2σ2) PDF of Standard Normal Distribution = P(x) = (1/sqrt(2π))e-(x2 / 2)
###### where,

m = Mean. σ = Standard Deviation. π = 3.14 e = 2.718

##### Example 1 :

Find Probability Density Function with, mean m=5 Standard deviation σ=2 Normal random variable x=10

###### Step 1:

To calculate PDF find sqrt(2π). sqrt(2π) = sqrt(2 x 3.14) = sqrt(6.28) = 2.51

###### Step 2:

Find 1/(σsqrt(2π)). σsqrt(2π) = 2 x 2.51 = 5.02 1/(σsqrt(2π)) = 1/5.02 = 0.199

###### Step 3:

To Find e-(x-m)2 / (2σ2) calculate -(x-m)2 and 2σ2. -(x-m)2 = -(10-5)2 = 52 = 25 2σ2 = 2 x (22) = 2 x 4 = 8 -(x-m)2 / (2σ2) = 25/8 = 3.125

###### Step 4:

Calculate e-(x-m)2 / (2σ2) = 2.7183.125 = 22.75

###### Step 5:

To find PDF formula is used. = 0.199 x 22.75 = 4.53

##### Example 2 :

Find Standard Normal Distribution(m=0; σ=1) with, Normal random variable x=2

###### Step 1:

Find 1/sqrt(2π). sqrt(2π) = 2.51 1/sqrt(2π)) = 1/2.51 = 0.398406375

###### Step 2:

Calculate e-(x2 / 2). (x2 / 2)= 22/2 = 2 e-(x2 / 2)= 2.718(-2) = 0.13534

###### Step 3:

To find Standard Normal Distribution Formula is used. = 0.398406375 x0.13534 = 0.0539

#### Related Articles:

This tutorial will help you to calculate the Probability Density Function(PDF) and Standard Normal Distribution.