Grinstead-Snell-Probability · Problem 6.3.12

When two random variables are independent, their joint expectation factors beautifully: $E(XY) = E(X) \cdot E(Y)$. Since both $X$ and $Y$ are uniform on $[0,1]$, each has expectation $\frac{1}{2}$—the midpoint of their range. Multiply these together to get your answer, then run a quick simulation by sampling pairs of uniform random numbers, computing their products, and averaging to confirm the theory matches reality.

When two random variables are independent, their joint expectation factors beautifully: $E(XY) = E(X) \cdot E(Y)$. Since both $X$ and $Y$ are uniform on $[0,1]$, each has expectation $\frac{1}{2}$—the midpoint of their range. Multiply these together to get your answer, then run a quick simulation by sampling pairs of uniform random numbers, computing their products, and averaging to confirm the theory matches reality.