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Let y₁,..., yn be a sample from a Poisson distribution with mean λ, where A is given a Gamma(a, ß) prior distribution. (a) It is observed that y₁ = y2 = yn = 0, and we take a = 1,8 = 1. i. What is the posterior distribution for X? ii. What is the posterior mean? iii. What is the posterior median and an equal tail 95% credible interval for X (without using R)? (b) Show that if a new data-point x is generated from the same Poisson distribution, the posterior predictive probability that x = = 0 is p(x = 0 | y) = = n+1 n+2 (c) Now suppose that we have general y₁,..., yn, a and B; and that again x is a new data-point from the same Poisson distribution. i. Find the mean and variance of x. ii. Derive the full posterior predictive distribution for x. (d) Use R to check as many of these results as you can.

Let y₁,..., yn be a sample from a Poisson distribution with mean λ, where A is given a Gamma(a, ß) prior distribution. (a) It is observed that y₁ = y2 = yn = 0, and we take a = 1,8 = 1. i. What is the posterior distribution for X? ii. What is the posterior mean? iii. What is the posterior median and an equal tail 95% credible interval for X (without using R)? (b) Show that if a new data-point x is generated from the same Poisson distribution, the posterior predictive probability that x = = 0 is p(x = 0 | y) = = n+1 n+2 (c) Now suppose that we have general y₁,..., yn, a and B; and that again x is a new data-point from the same Poisson distribution. i. Find the mean and variance of x. ii. Derive the full posterior predictive distribution for x. (d) Use R to check as many of these results as you can.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 19PFA
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please answer part d only asap using R and show the R code. will give you good feedback

Let y₁,..., yn be a sample from a Poisson distribution with mean λ, where A is given a Gamma(a, ß) prior distribution. (a) It is observed that y₁ = y2 = = Yn = 0, and we take a = 1, ß = 1. —= i. What is the posterior distribution for X? ii. What is the posterior mean? iii. What is the posterior median and an equal tail 95% credible interval for X (without using R)? (b) Show that if a new data-point x is generated from the same Poisson distribution, the posterior predictive probability that x = 0 is p(x=0|y) = n+1 n+2' (c) Now suppose that we have general y₁,..., Yn, a and ß; and that again x is a new data-point from the same Poisson distribution. i. Find the mean and variance of x. ii. Derive the full posterior predictive distribution for x. (d) Use R to check as many of these results as you can.
Transcribed Image Text:Let y₁,..., yn be a sample from a Poisson distribution with mean λ, where A is given a Gamma(a, ß) prior distribution. (a) It is observed that y₁ = y2 = = Yn = 0, and we take a = 1, ß = 1. —= i. What is the posterior distribution for X? ii. What is the posterior mean? iii. What is the posterior median and an equal tail 95% credible interval for X (without using R)? (b) Show that if a new data-point x is generated from the same Poisson distribution, the posterior predictive probability that x = 0 is p(x=0|y) = n+1 n+2' (c) Now suppose that we have general y₁,..., Yn, a and ß; and that again x is a new data-point from the same Poisson distribution. i. Find the mean and variance of x. ii. Derive the full posterior predictive distribution for x. (d) Use R to check as many of these results as you can.
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