Question

If X is a binomial random variable based on n trials and success probability p, and p has the conjugate beta prior with parameters alpha = 1 and eta = 1, (a) Determine the Bayes estimator for p, ˆpB . (b) What is another name for the beta distribution with alpha = 1 and eta = 1? (c) Find the mean square error (MSE) of the Bayes estimator found in part (a). d) For what values of p is the MSE of the Bayes estimator smaller than that of the unbiased estimator ˆp = X/n?

Name: if x is a binomial random variable based on n trials and success probability p and p has the conjugate beta prior with parameters alpha 1 and beta 1 a determine the bayes estimator for p pb 61185
Uploaded: 2022-11-10T02:31:06-08:00
Channel: Victor Salazar
Description: if x is a binomial random variable based on n trials and success probability p and p has the conjugate beta prior with parameters alpha 1 and beta 1 a determine the bayes estimator for p pb 61185

          If X is a binomial random variable based on n trials and success probability p, and p has the
conjugate beta prior with parameters alpha  = 1 and eta  = 1,
(a) Determine the Bayes estimator for p, ˆpB .
(b) What is another name for the beta distribution with alpha  = 1 and eta  = 1?
(c) Find the mean square error (MSE) of the Bayes estimator found in part (a). 
d) For what values of p is the MSE of the Bayes estimator smaller than that of the unbiased estimator ˆp = X/n?

Added by Marina V.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Victor Salazar

11/10/2022

Step 1

The Bayes estimator for a parameter p under squared error loss is the posterior mean. Given a beta prior with parameters α = 1 and β = 1, and a binomial likelihood, the posterior distribution is also a beta distribution with parameters α' = α + X and β' = β + n - Show more…

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If X is a binomial random variable based on n trials and success probability p, and p has the conjugate beta prior with parameters alpha = 1 and eta = 1, (a) Determine the Bayes estimator for p, ˆpB . (b) What is another name for the beta distribution with alpha = 1 and eta = 1? (c) Find the mean square error (MSE) of the Bayes estimator found in part (a). d) For what values of p is the MSE of the Bayes estimator smaller than that of the unbiased estimator ˆp = X/n?