Question 3.21: Suppose that X and Y are independent continuous random varia......

Conditioning on the value of Y yields

P\{X\lt Y\}=\int_{-\infty}^{\infty}P\{X\lt Y|Y=y\}\!f_{Y}(y)\;d y

=\int_{-\infty}^{\infty}P\{X\lt y|Y=y\}\!f_{Y}(y)\;d y

=\int_{-\infty}^{\infty}P\{X\lt y\}f_{Y}(y)\;d y

=\int_{-\infty}^{\infty}F_{X}(y)f_{Y}(y)\ d y

where

F_{X}(y)=\int_{-\infty}^{y}f_{X}(x)\;d x