Mean gap between two uniforms on three to four

Question

Two independent random variables $X$ and $Y$ are each uniform on the interval $(3,4)$ . Compute the expected absolute difference $E|X-Y|$ .

Short answer

$1/3$ . Shifting both by $3$ changes nothing, so this is $E|U_1-U_2|$ for two standard uniforms. The two points cut $[0,1]$ into three pieces that are exchangeable, so each, including the middle piece of length $|U_1-U_2|$ , has mean $\tfrac13$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Continuous Random Variables and Probability Density Functions

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