Question 4 [17 marks] The chi-squared distribution with parameter n > 0 is denoted as ma. If X ~ xm, then X has probability density function (pdf) i le i, r>0, fx(x) = 0 otherwise. a. Show that the k-th raw moment E(X*) = 21 (4+ 7) k = 1, 2, ... . b. Hence, show that the mean, variance, skewness and kurtosis of the x, distribution are E(X) = n, Var(X) = 2n, 53 = V8. S4 = 3 + -. c. Judging from ( and C, what would you say happens to the shape of the x? pdf for large n? d. Consider two independent chi-squared random variables U ~ xm and V ~ x2 Using the the convolution method, show that the distribution of their sum W = U + V is chi squared with parameter m + n: W Xmen