Read through the following questions and answer them approximately.

Question 5.11 A random variable X has a Gamma distribution with parameters o and 1. (i) Show that the moment generating function of X, My (t), is given by: M x (1) = ( 1-4 ) , 152 [3] (ii) Use this moment generating function to show that the mean and variance of X are given by 7 and , respectively. [2] (iii) In the case when 20 is an integer, prove that 21X is a X2, random variable. [3] [Total 8] Question 5.12 (i) Define the size and power of a statistical test. [2] An actuary is comparing a set of observed mortality rates with the values predicted by a certain formula. He has calculated the "standardised residuals" for 25 different ages. If the formula is satisfactory these residuals should have a standard normal distribution. He is assuming that the residuals X1..... X25 are independent identically distributed random variables from a N(#,1) distribution and he wishes to carry out a test of the hypotheses Ho : / =0 versus H :/=0. (ii) Write down the test criterion the actuary should use for a symmetrical test of size 5%. [2] (iii) Calculate the power of this test if the true value of / is: (a) 0 (b) 0.5 (c) -0.2 [3] (iv) Hence sketch a graph of the power function for this test as a function of M. [2] [Total 9]