1 f(y|a, B) r(a)3a 0 < y < ∞,a > 0, B > 0. (a) Prove that the moment generating function (mgf) of Y is My (t) = (1 – Bt)-a. (b) Let Y1,..., Y, be iid (independent and identically distributed) gamma(a, B) ran- dom variables. Derive the mgf of T = Y1 + ..+ Y, and hence the pdf of T. %3D
1 f(y|a, B) r(a)3a 0 < y < ∞,a > 0, B > 0. (a) Prove that the moment generating function (mgf) of Y is My (t) = (1 – Bt)-a. (b) Let Y1,..., Y, be iid (independent and identically distributed) gamma(a, B) ran- dom variables. Derive the mgf of T = Y1 + ..+ Y, and hence the pdf of T. %3D
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Suppose a random variable Y has gamma(a, B) distribution with pdf
f(y|a, B) =
1
rev/Bya-1,
0 < y <0, a > 0, B > 0.
(a) Prove that the moment generating function (mgf) of Y is My (t) = (1 – Bt)-ª.
(b) Let Y1,..., Y, be iid (independent and identically distributed) gamma(a, B) ran-
dom variables. Derive the mgf of T = Y1 + ..+ Yn and hence the pdf of T.
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