.7 In this problem we refer to Example 8.6. Suppose the random variables (X₁, Y₁), (X2, Y2),..., (Xn, re all independent and that for each i we have X₁, Yį distributed as N(μ₁,0²). a) Set v = and express the likelihood function in terms of the parameters H1,Hn, and v. b) Show that the log likelihood is

1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).

4. Estimation based on a function of the observation. Let e be a positive random variable, with known mean p and variance o?, to be estimated on the basis of a measurement X of the form X = vOW. We assume that W is independent of e with zero mean, unit variance, and known fourth moment E[W4]. Thus, the conditional mean and variance of X given O are 0 and 0, respectively, so we are essentially trying to estimate the variance of X given an observed value. (a) Find the linear LMS estimator of e based on X = x. (b) Let Y = X?. Find the linear LMS estimator of O based on Y = y.

Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.