Solution Found!
Let Y be the sum of the observations of a random sample
Chapter 6, Problem 1E(choose chapter or problem)
Problem 1E
Let Y be the sum of the observations of a random sample from a Poisson distribution with mean θ. Let the prior pdf of θ be gamma with parameters α and β.
(a) Find the posterior pdf of θ, given that Y = y.
(b) If the loss function is [w(y) − θ]2, find the Bayesian point estimate w(y).
(c) Show that w(y) found in (b) is a weighted average of the maximum likelihood estimate y/n and the prior mean αβ, with respective weights of n/(n + 1/β) and (1/β)/(n + 1/β).
Questions & Answers
QUESTION:
Problem 1E
Let Y be the sum of the observations of a random sample from a Poisson distribution with mean θ. Let the prior pdf of θ be gamma with parameters α and β.
(a) Find the posterior pdf of θ, given that Y = y.
(b) If the loss function is [w(y) − θ]2, find the Bayesian point estimate w(y).
(c) Show that w(y) found in (b) is a weighted average of the maximum likelihood estimate y/n and the prior mean αβ, with respective weights of n/(n + 1/β) and (1/β)/(n + 1/β).
ANSWER:
Step 1 of 6
Given:
Let Y be the sum of the observations of a random sample from a Poisson distribution with mean 