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Question 1 A six-sided die has an unknown number of faces marked with a six. Let k be this unknown number, which we would like to estimate. Our prior distribution for k is P(k=j)= (5/8, j = 1 1/16, j = 0,2,3,4,5,6. When the die is thrown each face has an equal chance of showing. The observed data is that the die was thrown twice, and it showed a six exactly once. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for k? (b) Derive the normalized posterior distribution for k. What is the posterior mean for k? (c) Find the posterior predictive probability that if the die is thrown again, it will not show a six.

Question 1 A six-sided die has an unknown number of faces marked with a six. Let k be this unknown number, which we would like to estimate. Our prior distribution for k is P(k=j)= (5/8, j = 1 1/16, j = 0,2,3,4,5,6. When the die is thrown each face has an equal chance of showing. The observed data is that the die was thrown twice, and it showed a six exactly once. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for k? (b) Derive the normalized posterior distribution for k. What is the posterior mean for k? (c) Find the posterior predictive probability that if the die is thrown again, it will not show a six.

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
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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Question 1 A six-sided die has an unknown number of faces marked with a six. Let k be this unknown number, which we would like to estimate. Our prior distribution for k is (5/8, j = 1 1/16, j = 0,2,3,4,5,6. P(k = j) = { When the die is thrown each face has an equal chance of showing. The observed data is that the die was thrown twice, and it showed a six exactly once. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for k? (b) Derive the normalized posterior distribution for k. What is the posterior mean for k? (c) Find the posterior predictive probability that if the die is thrown again, it will not show a six.
Transcribed Image Text:Question 1 A six-sided die has an unknown number of faces marked with a six. Let k be this unknown number, which we would like to estimate. Our prior distribution for k is (5/8, j = 1 1/16, j = 0,2,3,4,5,6. P(k = j) = { When the die is thrown each face has an equal chance of showing. The observed data is that the die was thrown twice, and it showed a six exactly once. (a) Write down the likelihood for the observed data. What is the maximum likelihood estimate for k? (b) Derive the normalized posterior distribution for k. What is the posterior mean for k? (c) Find the posterior predictive probability that if the die is thrown again, it will not show a six.
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