1. Recall that for each t > 0, the gamma function is defined, - [²² xt-le-dx. r(t) = (a) In class, we used the fact that T(a+1) r(a) = Show that this is indeed true by showing that for any a > 0, г(a + 1) Hint: Use integration by parts. = a to find the mean of the gamma distribution. al(a). (b) Suppose X~ Gamma(a, 3). Find E(X2) by "integrating without integrating". Use this and the formula we derived in class for the mean to find Var(X). (c) Suppose X₁, X2, ..., Xn are i.i.d. (independent and identically distributed) random variables with the Expo(A) distribution. Use moment generating functions to find the distribution of X₁ + ··· + Xn. Name this distribution and give the value of its parameter(s).

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