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B2. (a) Consider X₁,..., X₁ to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with = 0, 1, 2, 3,..., and pe (0, 1]. (i) Using the MGF (M(t) = 1-(1-p)e derive E[X] and Var[X]. (ii) Find the Maximum Likelihood Estimator (MLE) for p.

B2. (a) Consider X₁,..., X₁ to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with = 0, 1, 2, 3,..., and pe (0, 1]. (i) Using the MGF (M(t) = 1-(1-p)e derive E[X] and Var[X]. (ii) Find the Maximum Likelihood Estimator (MLE) for p.

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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B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., and p € (0, 1]. (i) Using the MGF (M(t) == (ii) Find the Maximum Likelihood Estimator (MLE) for p. -(1-p)et (b) Suppose X₁,..., X₁ is a random sample from a Beta(01, 1) population, and Y₁,..., Ym is an independent random sample from a Beta(02, 1) population. We want to find the approximate Likelihood Ratio Test for Ho: 01 02 00, versus H₁: 01 02. To this aim: = = (i) Under the alternative hypothesis H₁0₁02, show that the MLE for ₁ and 02 are: 0₁ 02 derive E[X] and Var[X]. n Σlog(x)' Recall, that the PDF of Beta(a, b) is fy (y) 00 = - m Σ log(yi) = ['(a) = (a − 1)! for all positive integers a and I (1) = 1) [(a+b) F(a) (b)-1(1-y)b-1 and (ii) Under the null hypothesis Ho: 0₁ 02 = 0o, show that the MLE for 00 is: 01 = n + m E log(xi) + log(yi)
Transcribed Image Text:B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., and p € (0, 1]. (i) Using the MGF (M(t) == (ii) Find the Maximum Likelihood Estimator (MLE) for p. -(1-p)et (b) Suppose X₁,..., X₁ is a random sample from a Beta(01, 1) population, and Y₁,..., Ym is an independent random sample from a Beta(02, 1) population. We want to find the approximate Likelihood Ratio Test for Ho: 01 02 00, versus H₁: 01 02. To this aim: = = (i) Under the alternative hypothesis H₁0₁02, show that the MLE for ₁ and 02 are: 0₁ 02 derive E[X] and Var[X]. n Σlog(x)' Recall, that the PDF of Beta(a, b) is fy (y) 00 = - m Σ log(yi) = ['(a) = (a − 1)! for all positive integers a and I (1) = 1) [(a+b) F(a) (b)-1(1-y)b-1 and (ii) Under the null hypothesis Ho: 0₁ 02 = 0o, show that the MLE for 00 is: 01 = n + m E log(xi) + log(yi)
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