Conditional probability mass function

Summary

The conditional probability mass function of Y given X

• \(X,Y\) are two discrete random variables with joint probability mass function \(f(x,y)\) and marginal probability mass functions \(f_X(x)\) and \(f_Y\left( y \right)\) .
• Define events \(A={Y=y}\) and \(B=\{ X=x \}\) . We have \(P\left( A \right)=f_Y\left( y \right)\) and \(P\left( B \right)=f_X\left( x \right)\) .
• The conditional probability \(P\left( B \right)\) is denoted

\[P\left( X=x \right)\]

• If \(P\left( B \right)≠0\) , the conditional probability is given by

\[P\left( B \right)= \frac{P\left( A∩B \right)}{P(B)}\]

• or

\[P\left( X=x \right)= \frac{f\left( x,y \right)}{f_X\left( x \right)}\]

• The conditional probability mass function of \(Y\) given \(X\) is defined as

\[f_{Y|X}\left( x \right)=P\left( X=x \right)= \frac{f\left( x,y \right)}{f_X\left( x \right)}\]

Example

\(f\left( x,y \right), f_X\left( x \right),f_Y(y)\) are defined by the following table:

• Then, for example, \(f_{Y|X}\left( 0 \right)=f(1,0)/f_X(0)=0.3/0.4=0.75\) . The probability that \(Y\) takes the value \(1\) if we know that \(X\) takes the value \(0\) is 0. 7 5.

The conditional probability mass function of X given Y

• The conditional probability mass function of \(X\) given \(Y\) is defined similarly

\[f_{X|Y}\left( y \right)=P\left( Y=y \right)= \frac{f\left( x,y \right)}{f_Y\left( y \right)}\]