Asked Aug 8, 2019

Suppose that Xi ∼ Gamma(αi , β) independently for i = 1, . . . , N. The mgf(moment generating function) of Xiis MXi(t) = (1 − (t/β) )−αi . (a)Use the mgf of Xi to derive the mgf of ∑i=1 Xi .

Use the mgf of Xi to derive the mgf of ∑i=1 Xi


Expert Answer

Step 1


It is given that Xi’s (i = 1, 2, …, N) are independent Gamma (αi, β) random variables with mgf as follows:


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M0- 1 B

Step 2


The mgf of the sum of all Xi’s, that is, of Σ Xi is calculated as follows:


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M()M+Xx ,(e) =Mx (t) My (t).. .M(t) X, are independent] B -(a+a2+..ay = mgf of Gamma (>a.B)

Step 3


The above derivation shows that Σ Xi has the Gamma (Σ α...

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