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.

Show that this defines a probability density. 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.