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{"ops":[{"insert":"Suppose that the continuous random variable has a Gamma distribution with shape parameter \u03b1 and scale parameter \u03b2, that is X \u223c \u0393(\u03b1, \u03b2) ."},{"insert":"\n"},{"insert":""},{"insert":"\n"},{"insert":"(a) Show that the moment generating function of the random variable X is Mx(t) = \u0012(1 \u2212t\/\u03b2)^\u2212\u03b1 ,"},{"insert":"\n"},{"insert":""},{"insert":"\n"},{"insert":"where t < \u03b2 and use the result to derive the expectation and variance of X."},{"insert":"\n"},{"insert":""},{"insert":"\n"},{"insert":"(b) Hence show that if (x1, \u00b7 \u00b7 \u00b7 , xn) are independent random variable such that each has a Gamma distribution with parameter P \u03b1i and \u03b2 where i = 1, 2, 3, \u00b7 \u00b7 \u00b7 , n , then the random variable Y =sum n, i=1 Xi , has a Gamma distribution with parameter Sum n, i=1 \u03b1i and \u03b2."},{"insert":"\n"},{"insert":""},{"insert":"\n"},{"insert":"(c) Deduce from your results in (ii) that if (x1, x2, \u00b7 \u00b7 \u00b7 , xn) are iid exponential distributed with"},{"insert":"\n"},{"insert":""},{"insert":"\n"},{"insert":"parameter \u03b2, then Sn = Sum Xi \u223c \u0393(n, \u03b2)."},{"insert":"\n"}]}

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