3. [22 marks] Let X = (X1, X2, ..., Xn) be i.i.d. random variables, each with a density f(x, 0) = 1 e(-10-PP), 1 > 0 0 elsewhere where 0 > 0 is a parameter. (This is called the log-normal density.) a) Prove that the family L(X, 0) has a monotone likelihood ratio in T = El= (In X;)?. b) Argue that there is a uniformly most powerful (UMP) a-size test of the hypothesis Ho : 0 Go and exhibit its structure. c) Using the density transformation formula (or otherwise) show that Y. = In Xi has a N(0, 02) distribution. d) Using c) (or otherwise), find the threshold constant in the test and hence determine completely the uniformly most powerful a- size test * of Ho : 0 60- Calculate the power function Ed* and sketch a graph of it