Let x be a single observation of an explambda random

Question

Let X be a single observation of an Exp(λ) random variable, which has pdf λ(x) =λ exp(-λx) if x ≥ 0, 0 if x <0.

Consider testing H0 : λ ≥ λ0 versus H1 : λ<  λ0.

(a) Find the power function of the hypothesis test that rejects H0 if and only if X ≥ c.

(b) Let 0 <α<  1. Find a value of c such that the test in part (a) has size α.

(c) For what true values of λ is Pλ(type II error) ≥ 1/2 for the test in part (a) with size α as in (b)?