B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, withprobability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., andp € (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,..., Ymis an independent random sample from a Beta(02, 1) population. We want to findthe 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 and02 are:0₁02derive E[X] and Var[X].nΣlog(x)'Recall, that the PDF of Beta(a, b) is fy (y)00= -mΣ 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 + mE log(xi) + log(yi)

The discrete probability distribution is the distribution that has variables that take an integer…

Answered: B2. (a) Consider X₁,..., X₁ to be a…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

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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.