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.

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)} = 2.718^3.125 = 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