Mean area of a right triangle with random legs

Question

Let $X$ and $Y$ be independent, each uniform on $\{1,2,3,4,5\}$ . Find the expected area of the triangle whose corners are $(0,0)$ , $(X,0)$ and $(0,Y)$ .

Short answer

$\tfrac92$ . The area is $\tfrac12XY$ , and by independence $\mathbb{E}[XY]=\mathbb{E}[X]\,\mathbb{E}[Y]=3\cdot3$ , giving $\tfrac92$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

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