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Suppose the random variable, X, follows a Bernoulli distribution with parameter 0. The probability function of X is given by -_-{oa- p(x; 0) = 0² (1-0)¹ if x = 0, 1, otherwise. Let X₁, X₂, X, be a random sample of size n from the population of X. Assume that the X's are independent and identically distributed with the same parameter. (a) Based on the joint probability function of the sample, state the likelihood function of the parameter. Discuss the difference between the joint probability function and the likelihood function. (b) Show that the sample mean is the maximum likelihood estimator (mle) of 0. (c) Find the method of moments estimator (mme) of 0.

Suppose the random variable, X, follows a Bernoulli distribution with parameter 0. The probability function of X is given by -_-{oa- p(x; 0) = 0² (1-0)¹ if x = 0, 1, otherwise. Let X₁, X₂, X, be a random sample of size n from the population of X. Assume that the X's are independent and identically distributed with the same parameter. (a) Based on the joint probability function of the sample, state the likelihood function of the parameter. Discuss the difference between the joint probability function and the likelihood function. (b) Show that the sample mean is the maximum likelihood estimator (mle) of 0. (c) Find the method of moments estimator (mme) of 0.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Suppose the random variable, X, follows a Bernoulli distribution with parameter 0. The probability function of X is given by { [ 0² (1 - 0)¹-² if x = 0,1, otherwise. p(x; 0) = Let X₁, X₂, X be a random sample of size n from the population of X. Assume that the X's are independent and identically distributed with the same parameter. (a) Based on the joint probability function of the sample, state the likelihood function of the parameter. Discuss the difference between the joint probability function and the likelihood function. (b) Show that the sample mean is the maximum likelihood estimator (mle) of 0. (c) Find the method of moments estimator (mme) of 0.
Transcribed Image Text:Suppose the random variable, X, follows a Bernoulli distribution with parameter 0. The probability function of X is given by { [ 0² (1 - 0)¹-² if x = 0,1, otherwise. p(x; 0) = Let X₁, X₂, X be a random sample of size n from the population of X. Assume that the X's are independent and identically distributed with the same parameter. (a) Based on the joint probability function of the sample, state the likelihood function of the parameter. Discuss the difference between the joint probability function and the likelihood function. (b) Show that the sample mean is the maximum likelihood estimator (mle) of 0. (c) Find the method of moments estimator (mme) of 0.
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