6.32 The random variable X has an extreme-value distribution, with cumulative distribution function:

P(X ≤ x)=1−

k x

a

, with parameters a and k, for a > 0, x ≥ k > 0.

Explain how to simulate X using the inversion method.

6.33 The random variable X has a logistic distribution, with pdf given by f(x) = e−x

(1 + e−x)2 for −∞≤ x ≤ ∞.

(i) Use the inversion method to construct an algorithm to simulate X.

(ii) If U1, U2 are independent U(0, 1) random variables, let X+ = − loge U1 if U2 < (1 + U1)−2. Show that this results from a rejection method for simulating X, based on an exponential envelope. Explain how you would modify X+ to simulate X.