# Conditional expectation $E[X|Y<y]$

Let $X:\Omega \to \mathbb{R}$ and $Y:\Omega \to \mathbb{R}$. Consider the joint pdf $f_{XY}(x,y)$ and univariate pdfs $f_X(x)$ and $f_Y(y)$.

Is it true that $E[X|Y < y]$ equals:

$$\displaystyle \int_{-\infty}^\infty \int_{-\infty}^y x \frac{f_{XY}(x,y)} {f_{Y}(y)} dydx$$

since $f_{XY}(x | y) = \frac{f_{XY}(x,y)} {f_{Y}(y)}$.

• You need to be a little careful with notation. Does $f_Y^{-1}(y)$ mean the inverse function of $f_Y$ or $\frac{1}{f_Y(y)}$? Apr 17, 2013 at 0:46
• @DilipSarwate Good point, clarification added. Apr 17, 2013 at 1:01
• I guess the $dydx$ at the end should be $dxdy$ or that the upper limit of the inner integral is $x$ or that...
– Did
Apr 17, 2013 at 2:52

$$E[X\mid Y\lt y]=\frac1{F_Y(y)}\,\int_{-\infty}^{+\infty}x\int_{-\infty}^yf_{X,Y}(x,t)\,\mathrm dt\,\mathrm dx.$$