# Question: Let X have an exponential distribution with 1

Let X have an exponential distribution with θ = 1; that is, the pdf of X is f(x) = e−x, 0 < x < ∞. Let T be defined by T = ln X, so that the cdf of T is

G(t) = P(ln X ≤ t) = P(X ≤ et)

(a) Show that the pdf of T is g(t) = ete−et, −∞ < x < ∞, which is the pdf of an extreme-value distribution.

(b) Let W be defined by T = α + β ln W, where −∞ < α < ∞ and β > 0. Show that W has a Weibull distribution.

G(t) = P(ln X ≤ t) = P(X ≤ et)

(a) Show that the pdf of T is g(t) = ete−et, −∞ < x < ∞, which is the pdf of an extreme-value distribution.

(b) Let W be defined by T = α + β ln W, where −∞ < α < ∞ and β > 0. Show that W has a Weibull distribution.

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