A random variable X has the Weibull distribution W

(b,

c) if its density is c

b

x b

c−1 e

−(x/b)c

, x> 0,

b, c > 0.

(i) Show that this defines a probability density.

(ii) If X1,...,Xn is a sample from W

(b, c), with the shape parameter c known, show that there exists a UMP test of H : b ≤ b0 against b>b0 and give its form.