Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 7 / b / iii

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- Solution iii

Solution

For

W = max {U, V}

and

0 \leq w \leq 1

, independence gives the cumulative distribution function

F_{W} (w) = P (U \leq w, V \leq w) = w^{2} .

The cumulative distribution function is zero below

0

and one above

1

. Differentiating on the interior gives

f_{W} (w) = 2 w (0 < w < 1), f_{W} (w) = 0 otherwise,

up to irrelevant endpoint values. There are no endpoint atoms. This maximum is an order statistic.

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