Let [math]X[/math] and [math]Y[/math] be independent random variables with uniform density functions on [math][0,1][/math]. Find [math]E(|X - Y|)[/math].

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: B

We condition on [math]Y=y[/math] then the expected value of [math]|X-y|[/math] equals

[[math]] \int_{0}^y y-x \, dx + \int_{y}^1 x-y \, dx = \int_{0}^{y} w \, dw + \int_{0}^{1-y} w \, dw = \frac{y^2}{2} + \frac{(1-y)^2}{2}. [[/math]]

Hence the expected value of |X-Y| equals

[[math]] \int_{0}^1 \frac{y^2}{2} + \frac{(1-y)^2}{2} \, dy = 2\int_{0}^1 \frac{y^2}{2} dy = \frac{1}{3}. [[/math]]