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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) (10 pts) Find the MLE of p. (b) (5 pts) Find the estimator for p using method of moments.

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) (10 pts) Find the MLE of p. (b) (5 pts) Find the estimator for p using method of moments.

MATLAB: An Introduction with Applications
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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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) (10 pts) Find the MLE of p. (b) (5 pts) Find the estimator for p using method of moments. (c) (5 pts) Now let's think like a Bayesian. Consider a Beta prior on p, i.e., p ~ Beta(@, ß). Find the posterior distribution of p. Hint: For Geometric likelihood, the conjugate prior on p is a Beta distribution. (d) (5 pts) What is the Bayes estimator of p under squared error loss? Denote it by Pg. (e) (5 pts) What happens to PB if both a and 3 goes to 0?
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) (10 pts) Find the MLE of p. (b) (5 pts) Find the estimator for p using method of moments. (c) (5 pts) Now let's think like a Bayesian. Consider a Beta prior on p, i.e., p ~ Beta(@, ß). Find the posterior distribution of p. Hint: For Geometric likelihood, the conjugate prior on p is a Beta distribution. (d) (5 pts) What is the Bayes estimator of p under squared error loss? Denote it by Pg. (e) (5 pts) What happens to PB if both a and 3 goes to 0?
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