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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) :) 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?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter2: Exponential, Logarithmic, And Trigonometric Functions
Section2.CR: Chapter 2 Review
Problem 111CR: Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data...
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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) :) 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?
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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