Proposition (expected value of h(X,Y))

Let $X$ and $Y$ be jointly distributed rv’s with $\mathrm{pmf}p\left(x,y\right)$ or $\mathrm{pdf}f\left(x,y\right)$ according to whether the variables are discrete or continuous. Then the expected value of a function $h\left(X,Y\right)$, denoted by $E\left[h\left(X,Y\right)\right]$ or ${\mu }_{h\left(X,Y\right)}$, is given by

$$\{\begin{array}{ll}\sum _x\sum _{y}h\left(x,y\right)\cdot p\left(x,y\right)& \text{ if }X\text{ and }Y\text{ are discrete }\\ {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }h\left(x,y\right)\cdot f\left(x,y\right)dxdy& \text{ if }X\text{ and }Y\text{ are continuous }\end{array}$$