Computing Expected Values From Joint Distributions

Apply the law of the unconscious statistician to compute $E[g(X,Y)]$ directly from a joint PMF or joint PDF, without first finding marginal distributions.

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Tutorial

Expected Value of a Function of Two Discrete Random Variables

The expected value of a function $g(X, Y)$ of two jointly distributed discrete random variables is computed by weighting $g(x, y)$ by the joint PMF $p(x, y)$ and summing over all outcomes:

E[g(X, Y)] = \sum_{x} \sum_{y} g(x, y)\, p(x, y).

This formula handles any function of the two variables — sums, products, squares, indicators — without first having to find the marginal distributions.

For example, suppose $X \in \{0, 1\}$ and $Y \in \{1, 2\}$ with joint PMF

p(0,1) = 0.4,\quad p(0,2) = 0.1,\quad p(1,1) = 0.2,\quad p(1,2) = 0.3.

Then

\begin{align*} E[XY] &= (0)(1)(0.4) + (0)(2)(0.1) + (1)(1)(0.2) + (1)(2)(0.3) \\ &= 0 + 0 + 0.2 + 0.6 \\ &= 0.8. \end{align*}