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Problem 10*. Another property of the moment generating function is that if two random variables have
the same m.g.f., then they have the same p.d.f. (we say that the mapping p.d.f. I—) m.g.f. is one—to—one). In
class we showed that if we have X ~ Gamma(a,/\), then 1/)X(t) = (fl)? Use these facts and part (c) of
the previous problem to show that if X ~ Gamma(a, A) and Y N Gammafifl, A) are independent random variables and we define Z = X + Y, then Z ~ Gamma(a + £3, A).

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