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

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

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