(a) The random variables X1, . . . Xn are i.i.d. N (0, 1) random variables. The sample variance is defined as n S'2 n 1 E (Xi - X)2. i=1 In STAT2371 we shall show that (n - 1) S2 has a x2 distribution with n - 1 degrees of freedom. For the special case where n = 2, show this using both (i) the convolution formula, and (ii) the method of moment generating functions. (b) Let U1, U2, . .., Uk be i.i.d. Uniform(0, 1) random variables and X1, X2 be i.i.d. N (0, 1) random variables. Show that X? + X? has an exponential distribution with rate 1/2. Hence show that k W = -2log II Vi i=1 has the x2 distribution with 2k degrees of freedom. (c) The result in part (b) may be used to produce realizations of W. Implement this procedure in R to generate 1000 realizations of a chi-squared random variable with 10 degrees of freedom. Produce the histogram of the generated values w1, w2, ..., W1000. Find the mean and the variance of these values, and compare with their theoretical counterparts