10.2 The log-logistic distribution with the probability density function f(y) = e y 1 (1+e y ) 2 is sometimes used for

10.2 The log-logistic distribution with the probability density function f(y) = e

θ λy

λ−1

(1+e

θ y

λ )

2 is sometimes used for modelling survival times.

(a) Find the survivor function S(y), the hazard function h(y) and the cumulative hazard function H(y).

(b) Show that the median survival time is exp(−θ/λ).

(c) Plot the hazard function for λ = 1 and λ = 5 with θ = −5, θ = −2 and

θ =

1 2

.