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Suppose that X|A = 1 ~ Poisson(A) and that ^~ Gamma(a, B) (A priori distribution). Prove using the Total Probability Theorem that the unconditional density of X has a distribution Negative binomial of parameters r = a and p = 1+8· Hint: "Bring the integral to the gamma function".

Suppose that X|A = 1 ~ Poisson(A) and that ^~ Gamma(a, B) (A priori distribution). Prove using the Total Probability Theorem that the unconditional density of X has a distribution Negative binomial of parameters r = a and p = 1+8· Hint: "Bring the integral to the gamma function".

A First Course in Probability (10th Edition)
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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Suppose that X|^ = ^ ~ Poisson(A) and that A ~ Gamma(a, B) (A priori distribution). Prove using the Total Probability Theorem that the unconditional density of X has a distribution Negative binomial of parameters r = a and p = 1 1+B' Hint: "Bring the integral to the gamma function".
Transcribed Image Text:Suppose that X|^ = ^ ~ Poisson(A) and that A ~ Gamma(a, B) (A priori distribution). Prove using the Total Probability Theorem that the unconditional density of X has a distribution Negative binomial of parameters r = a and p = 1 1+B' Hint: "Bring the integral to the gamma function".
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