Exercise 3.15 (Extreme-Value Distribution) Consider a sequence of IID random variables Xi Exp(1/), i = 1,

Exercise 3.15 (Extreme-Value Distribution) Consider a sequence of IID random variables Xi ∼ Exp(1/θ), i = 1, 2, . . . , and define

The distribution function of Yn is denoted by Fn(x). Prove that

so that

This distribution is called a double exponential distribution and is known as one of the extreme-value distributions.

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Yn = max{X1, X2,..., Xn}-0logn.