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)
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Chapter1: Combinatorial Analysis
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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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