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, B = 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 3; 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.
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, B = 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 3; 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.
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 all parts asap showing clearly the methods. for part d please use basic r code (dont import any packages) and show the 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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Step 1: Write the given information.
VIEWStep 2: Determine the posterior distribution for λ.
VIEWStep 3: Determine the posterior mean and 95% credible interval for λ.
VIEWStep 4: Provide the proof for posterior predictive probability when new data point x is genrated.
VIEWStep 5: Determine the mean and variance of x.
VIEWStep 6: Derive the full posterior predictive distribution for x.
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