Maximum of four Unif(0,4) draws in (2,3]

Question

Four independent random variables are each uniform on $(0,4)$ . What is the probability that the largest of them lies in $(2,3]$ ?

Short answer

$65/256$ . The maximum is at most $x$ exactly when all four are, with probability $(x/4)^4$ . So the answer is $(3/4)^4-(2/4)^4=\tfrac{81-16}{256}$ .

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Lessons behind this question

Order Statistics and Sample Quantiles
Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions

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