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 . Determine the distribution of ∑i=1 Xi based on its mgf.

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 . Determine the distribution of ∑i=1 Xi based on its mgf.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.3: Rates Of Change

Problem 2E

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Question

Suppose that X_i ∼ Gamma(α_i , β) independently for i = 1, . . . , N. The mgf(moment generating function) of X_iis M_Xi(t) = (1 − (t/β) )^−αi . (a)Use the mgf of X_i to derive the mgf of ∑_i=1 X_i .

Determine the distribution of ∑_i=1 X_i based on its mgf.

Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.

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