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A box contains m = 5 balls, of which r are red and the rest black. The unknown quantity is r. Our prior distribution is that each value r = 0, 1,...,m has equal probability. We are told that twice, a ball was taken out and immediately replaced, and both times the ball was red. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for r? (b) Derive the normalized posterior distribution for r. What is the posterior mean for r? (c) Find the posterior predictive probability that if another ball is taken from the box, it is black.

A box contains m = 5 balls, of which r are red and the rest black. The unknown quantity is r. Our prior distribution is that each value r = 0, 1,...,m has equal probability. We are told that twice, a ball was taken out and immediately replaced, and both times the ball was red. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for r? (b) Derive the normalized posterior distribution for r. What is the posterior mean for r? (c) Find the posterior predictive probability that if another ball is taken from the box, it is black.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 8CR
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A box contains m = 5 balls, of which r are red and the rest black. The unknown quantity is r. Our prior distribution is that each value r = 0, 1,...,m has equal probability. We are told that twice, a ball was taken out and immediately replaced, and both times the ball was red. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for r? (b) Derive the normalized posterior distribution for r. What is the posterior mean for r? (c) Find the posterior predictive probability that if another ball is taken from the box, it is black.
Transcribed Image Text:A box contains m = 5 balls, of which r are red and the rest black. The unknown quantity is r. Our prior distribution is that each value r = 0, 1,...,m has equal probability. We are told that twice, a ball was taken out and immediately replaced, and both times the ball was red. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for r? (b) Derive the normalized posterior distribution for r. What is the posterior mean for r? (c) Find the posterior predictive probability that if another ball is taken from the box, it is black.
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