What is the error made if we approximate the probability

Question

1. Suppose that T has an exponential distribution with constant hazard 0.2.

(a) What is the probability that T‾ = 3?

(b) What is the error made if we approximate the probability that 2.5 ≤ T ≤ 2.75 by using the standard approximation?

2. Consider the distribution with density function f (t) =β²te^-βt, t ≥ 0. (This is a gamma distribution with first parameter 2.)

(a) Show that s(t) = (1 + βt)e^-βt

(b) Find the hazard function μ(t).

(c) Describe precisely the shifted distribution T ? u. (Hint: It is a mixture of two well-known distributions.)