The marginal probabillity mass function

Summary

• \(X\) and \(Y\) are two discrete random variables with joint probability mass function \(f(x,y)\) .
• The probability mass function of \(X\) , denoted by \(f_X(x)\) , is called the marginal probability mass function of \(X\) .
• The probability mass function of \(Y\) , denoted by \(f_Y(y)\) , is called the marginal probability mass function of \(Y\) .
• If the range of \(X\) is \({x_1,…,x_n}\) and range of \(Y\) is \({y_1,…,y_m}\) we have the following results

\[f_X\left( x_i \right)=\sum_{j=1}^{m}{ f\left( x_i,y_j \right) } for i=1,…,n\]

\[f_Y\left( y_j \right)=\sum_{i=1}^{n}{ f\left( x_i,y_j \right) } for j=1,…,m\]

Example

\(f(x,y)\) is defined by the following table

You get the marginal densities by summing vertically and horizontally