The Rule of the Lazy Statistician for Two Random Variables

Suppose $X$ and $Y$ are random variables with a known joint distribution, and we wish to compute the expected value of some function $g(X,Y).$ The Rule of the Lazy Statistician (LOTUS) lets us evaluate $E[g(X,Y)]$ directly from the joint distribution, without first deriving the distribution of the new random variable $Z=g(X,Y).$

In the discrete case, if $X$ and $Y$ have joint PMF $p_{X,Y}(x,y),$ then

$$E[g(X,Y)] = \sum\limits_{x}\sum\limits_{y} g(x,y)\, p_{X,Y}(x,y).$$

The double sum ranges over all $(x,y)$ pairs in the support.

For example, suppose $X$ and $Y$ have joint PMF

$$\begin{array}{c|cc} p_{X,Y}(x,y) & y=0 & y=1 \\ \hline x=0 & 0.1 & 0.2 \\ x=1 & 0.3 & 0.4 \end{array}$$

Then

$$\begin{align*} E[X+Y] &= \sum\limits_x\sum\limits_y (x+y)\, p_{X,Y}(x,y) \\ &= (0+0)(0.1) + (0+1)(0.2) \\ &\quad + (1+0)(0.3) + (1+1)(0.4) \\ &= 0 + 0.2 + 0.3 + 0.8 \\ &= 1.3. \end{align*}$$