X and Y are independent random variables, each with density f(t) = 2t for 0 ≤ t ≤ 1 and 0 elsewhere. Find E[|X − Y|]. Give the answer as a fraction in lowest terms.

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$X$ and $Y$ are independent random variables, each with density $f (t) = 2 t$ for $0 \leq t \leq 1$ and $0$ elsewhere. Find $E [∣ X - Y ∣]$ . Give the answer as a fraction in lowest terms.

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Hint

By symmetry, compute the expectation over the region $y < x$ and double it.

Solution

The joint density is $4 x y$ on the unit square, and the regions $y < x$ and $y > x$ contribute equally.
For fixed $x$ , the integral of $(x - y) \times 4 x y$ over $0 \leq y \leq x$ is $4 x (\frac{x ^{3}}{2} - \frac{x ^{3}}{3}) = \frac{2 x ^{4}}{3}$ .
Integrating over $0 \leq x \leq 1$ gives $\frac{2}{3} \times \frac{1}{5} = \frac{2}{15}$ .
Doubling for the other region, $E [∣ X - Y ∣] = 2 \times \frac{2}{15} = \frac{4}{15}$ .

Answer: $\frac{4}{15}$

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