hw1_sol

# The prior and the posterior are in the same family of

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Unformatted text preview: hyperparameters were set to a = 3, b = 1, ￿ 6 p(z |y ) ∼ Neg-Binomial 16, 7 6 ￿ 4. Suppose y1 , . . . , yn is a random sample from a negative binomial distribution with parameters (r, θ), where r is known and θ is unknown. Here, r denotes the number of successes and θ denotes the success probability. Suppose θ ∼ Beta(α, β ). (a) Derive the posterior distribution of θ. The likelihood function of θ is given by L ( θ ; y1 , . . . , y n ) = n ￿ i=1 ￿￿ ￿ ￿ ￿ ￿ n ￿ yi + r − 1 yi + r − 1 r θ (1 − θ)yi = yi yi i=1 ￿￿ θnr (1 − θ) ￿n i=1 yi The posterior distribution of θ is given by p(θ|y ) ∝ θnr (1 − θ) ￿n i=1 yi α − 1 θ = θα+nr−1 (1 − θ)β + Thus, θ|y ∼ Beta(α + nr, β + ￿n ￿n (1 − θ)β −1 i=1 yi − 1 i=1 yi ). The prior and the posterior are in the same family of distribution, so the beta prior is conjugate to the negative-binomial sampling model. (b) Construct a 95% credible interval for θ using α = β = 1. For the observed data, ￿10 i=1 yi = 75 and n...
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## This note was uploaded on 02/22/2013 for the course MATH 604 taught by Professor Tadasee during the Spring '13 term at Georgetown.

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