Only number 3 in the following picture:

._._._...._._.- _...._...,... -1 _-__. _. - Problem 3. Hypothesis test with a continuous observation 2/3 points (graded) Let 8 be a Bernoulli random variable that indicates which one of two hypotheses is true, and let P (0 = 1) = 19. Under the hypothesis ('3 = 0, the random variable X has a normal distribution with mean 0, and variance 1. Under the alternative hypothesis (9 = 1, X has a normal distribution with mean 2 and variance 1. Consider the MAP rule for deciding between the two hypotheses, given that X = m. 1. Suppose for this part of the problem thatp = 2/3. The MAP rule can choose in favor of the hypothesis 9 = 1 if and only if a: 2 c1. Find the value of c1. 61 = 0.6534 V/ 2. For this part, assume again that p : 2/3. Find the conditional probability of error for the MAP decision rule, given that the hypothesis 8 = 0 is true. 3. Find the overall (unconditional) probability of error associated with the MAP rule forp = 1/2. _x ._._._..._.__._._..l_. _' ____ __ __ Problem 5. Hypothesis test between two normals 2 points possible (graded) Conditioned on the result of an unbiased coin flip, the random variables T1,T2, . . . ,Tn are independent and identically distributed, each drawn from a common normal distribution with mean zero. If the result of the coin flip is Heads, this normal distribution has variance 1; otherwise, it has variance 4. Based on the observed values t1 , t2, . . . , tn, we use the MAP rule to decide whether the normal distribution from which they were drawn has variance 1 or variance 4. The MAP rule decides that the underlying normal distribution has variance 1 if and only if Find the values of c1 2 0 and c2 2 0 such that this is true. Express your answer in terms of n, and use "In" to denote the natural logarithm function, as in "ln(3)". c1 Problem 6. Determining the type of a lightbulb 3 points possible (graded) The lifetime of a typeA bulb is exponentially distributed with parameter A. The lifetime of a typeB bulb is exponentially distributed with parameter p, where ,u > A > 0. You have a box full of lightbulbs of the same type, and you would like to know whether they are of type A or B. Assume an a priori probability of 1/4 that the box contains typeB lightbulbs. 1. Assume that ,u. 2 3A. You observe the value t1 of the lifetime, T1, of a lightbulb. A MAP decision rule decides that the lightbulb is of type A if and only if t1 2 (1. Find 0:, and express your answer in terms of ,u, and A. Use 'mu" and 'Iambda" and 'ln" to denote u, A, and the natural logarithm function, respectively. For example, 111 _,u should be A entered as 'ln((2*mu)/lambda)". a: 2. Assume again that ,u, 2 3A. What is the probability of error of the MAP decision rule? O 1 7,1,: 3 7A _ (I _ 1 CI: 46 +4( 6 ) 3 1 O Zepa+ 4(1_e)\\a) 3. Assume that A = 3 and ,u. = 4. Find the LMS estimate of T2, the lifetime of another lightbulb from the same box, based on observing T1 = 2. Assume that conditioned on the bulb type, bulb lifetimes are independent. (For this part, you will need a calculator. Provide an answer with an accuracy of three decimal places.) LMS estimate of T2: We have k coins. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0, 1]. We assume that conditioned on Q = 9, all coin tosses are independent. Let T; be the number of tosses of the 13th coin until that coin results in Heads for the first time, for i = 1, 2,. . .,k. (T)- includes the toss that results in the first Heads.) You may find the following integral useful: For any non-negative integers k and m. kim! 1 f0 qk(l_q)mdq= (k-lm+ 1)!' 1. Find the PMF of T1. (Express your answer in terms oft using standard notation.) Fort = 1,2,..., pr. (1?) = 2. Find the least mean squares (LMS) estimate of Q based on the observed value, t, of T1. (Express your answer in terms of t using standard notation.) ElQlT1=tl= 3. We flip each of the k coins until they result in Heads for the first time. Compute the maximum a posteriori (MAP) estimate q\" of Q given the number of tosses needed, T1 = t1 , . . . ,Tk = tk, for each coin. Choose the correct expression for q\" 0 . 1:71 q k Zr=1ti O k g\" i. Zia ti 0 . k+1 q k Zi=1ti