Discussion

Suppose X and Y are independently and uniformly distributed random variables on the interval (0, 1). Then the distribution of max (X, Y) is :

Answer: D

max(X, Y) has the Beta(2, 1) distribution.

For \(0 \lt z \lt 1\),

\[ P(\max(X, Y) \le z) = P(X \le z)P(Y \le z) = z^{2} \]

so the density is \(2z\) on (0, 1). The Beta(a, b) density is proportional to \(z^{a-1}(1 - z)^{b-1}\), and \(2z\) matches \(a = 2\), \(b = 1\).

Beta(1, 2), with density \(2(1 - z)\), is the distribution of the minimum. Beta(1, 1) is the uniform, and Beta(2, 2) has density \(6z(1 - z)\).

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