The marginal probability density function

Summary

• \(X\) and \(Y\) are two continuous random variables with joint probability density function \(f(x,y)\) .
• The probability density function of \(X\) , denoted by \(f_X(x)\) , is called the marginal probability density function of \(X\) .
• The probability density function of \(Y\) , denoted by \(f_Y(y)\) , is called the marginal probability density function of \(Y\) .
• We have the following results

\[f_X\left( x \right)=\int_{-∞}^{∞}{ f\left( x,y \right)dy }\]

\[f_Y\left( y \right)=\int_{-∞}^{∞}{ f\left( x,y \right)dx }\]

Example

• \(f\left( x,y \right)=4xy\) for \(0≤x≤ 1\) and \(0≤y≤ 1\) . Then

\[f_X\left( x \right)=\int_{0}^{1}{ 4xydy }={\left[ 2xy^2 \right]}_0^1=2x\]

\[f_Y\left( y \right)=\int_{0}^{1}{ 4xydx }={\left[ 2x^2y \right]}_0^1=2y\]