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.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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please answer part d only asap using R and show the R code. will give you good feedback
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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