In this problem we refer to Example 8.6. Suppose the random variables (X₁, Y₁), (X2, Y2),..., (Xn, Yn)are all independent and that for each i we have X₁, Yį distributed as N(μi, 0²).a) Set v = and express the likelihood function in terms of the parameters1,,n, and v.b) Show that the log likelihood isn(Xì - Hi) + (Yi - Hi)21 (μ₁,..., fln, v) = 2n log. + n logv - v1√2πi=1c) Calculate the MLEs for the parameters 1,...,fln, V. EXAMPLE 8.6 The Neyman-Scott ParadoxHere is an example of a distribution where the MLE is not consistent. Youwill be asked to prove most of the claims here in Problem 8.7. Suppose randomvariables (X₁, Y₁), (X2, Y2),..., (Xn, Yn) are all independent and that for eachi we have X₁, Y₁ distributed as N(i, 02). Note that the variables have meansindexed by i but they all have the same variance. In this case, we can form thelog-likelihood function and show that the MLEs for the parameters are22==Xi + Yi2n1-n(Xi - Yi)²4These are in fact the unique solutions of the likelihood equation.Because of the last expression, it is easy to see that222and therefore the MLE is not consistent (it is biased).(8.4)(8.5)

We have two independent random variables Xi,Yi i=1,2,....,n and for each i, they are distributed as…

Answered: 8.7 In this problem we refer to Example…

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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If the moment generating function of a random variable X is: (1/3+(2/3)e t ) 5 find P (X > 3).
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Suppose that X is an exponential random variable with mean 5. (The cumulative distribution function is F(x) = 1- e-x/5 for x >= 0, and F(x) = 0 for x < 0. (a) Compute P(X > 5). (b) Compute P(1.4 <= X <= 4.2). (c) Compute P(1.4 < X < 4.2).
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The random variable x is known to be uniformly distributed between 1.0 and 1.5. B. Compute P(x=1.25) C. Compute P(1<=x<=1.25) D. Compute P(1.2<x<1.5)
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There are two traffic lights on a commuter's route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X1, X2 is a random sample of size n = 2). x1 0 1 2 p(x1) 0.1 0.2 0.7 ? = 1.6, ?2 = 0.44 (a) Determine the pmf of To = X1 + X2. to 0 1 2 3 4 p(to) (b) Calculate ?To. ?To = How does it relate to ?, the population mean? ?To = · ? (c) Calculate ?To2. ?To2 = How does it relate to ?2, the population variance? ?To2 = · ?2
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