Let X1,..., X, be a random sample (i.i.d.) from Geometric(p) distribution with PMF P(X = x) = (1 – p)*p, x = 0, 1, 2, .. The mean of this distribution is (1 – p)/p. (a) :) Find the MLE of p. (b) ( s) Find the estimator for p using method of moments. (c) Now let's think like a Bayesian. Consider a Beta prior on p, i.e., p ~ Beta(a, 3). Find the posterior distribution of p. Hint: For Geometric likelihood, the conjugate prior on p is a Beta distribution. (d) What is the Bayes estimator of p under squared error loss? Denote it by PB. (e) What happens to pg if both a and ß goes to 0?
Let X1,..., X, be a random sample (i.i.d.) from Geometric(p) distribution with PMF P(X = x) = (1 – p)*p, x = 0, 1, 2, .. The mean of this distribution is (1 – p)/p. (a) :) Find the MLE of p. (b) ( s) Find the estimator for p using method of moments. (c) Now let's think like a Bayesian. Consider a Beta prior on p, i.e., p ~ Beta(a, 3). Find the posterior distribution of p. Hint: For Geometric likelihood, the conjugate prior on p is a Beta distribution. (d) What is the Bayes estimator of p under squared error loss? Denote it by PB. (e) What happens to pg if both a and ß goes to 0?
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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Transcribed Image Text:Let X1, ... , Xn be a random sample (i.i.d.) from Geometric(p) distribution with PMF
P(X = x) = (1 – p)*p,
x = 0, 1, 2, . ..
The mean of this distribution is (1 – p)/p.
(a)
:) Find the MLE of p.
(b) (
s) Find the estimator for p using method of moments.
(c)
Now let's think like a Bayesian. Consider a Beta prior on p, i.e., p~ Beta(a, B).
Find the posterior distribution of p.
Hint: For Geometric likelihood, the conjugate prior on p is a Beta distribution.
(d)
) What is the Bayes estimator of p under squared error loss? Denote it by PB.
(e)
What happens to på if both a and ß goes to 0?
Expert Solution
Step 1
According to Bartleby guildlines we have to solve first three subparts and rest can be reposted....
We have given that
X follows Geometric distribution having the probability mass function is
P(X=x)=(1-p)xp. x=0,1,2....
And the mean of geometric distribution
E(X)=(1-p)/p
Then we have to MLE of p, Method of moment estimator of p and posterior distribution of p:
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