30 Dec 2018

Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

What is Conditional Probability Density Function (Conditional PDF)?

A probability density function is known as conditional PDF, when one random variable out of two random variables has a fixed value. Here suppose we have two random variables X and Y, and X has a fixed value equal to x. 
In this case in the conditional PDF of Y when X=x is given as-



In the same way, Conditional PDF of X when random variable Y takes a fixed value Y=y is given as-



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Properties of Conditional Probability Density Function (Conditional PDF)


Property 1- Conditional PDF is a non-negative function.



As conditional PDF is a ratio of two PDFs and we know that PDF is a non-negative function. Therefore the ratio of two non-negative PDFs is also non- negative function.

Property 2- Area under the conditional PDF is unity i.e. one.
This property mathematically can be expressed as-



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Property 3- Conditional probability density function (conditional PDF) reduces to marginal density if random variables X and Y are statistically independent.




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