課程名稱︰偵測與評估 課程性質︰選修 課程教師︰李枝宏 開課學院：電資 開課系所︰電信所/電機所 考試日期（年月日）︰ 考試時限（分鐘）：120mins 是否需發放獎勵金：Yes （如未明確表示，則不予發放） 試題 : Problem 1 (15%) Consider a binary detection problem: the null hypothesis H0 with probability P(H0) and the alternative hypothesis H1 with probability P(H1), let Cij be the cost of deciding Hi when Hj is acting. The conditional probability density functions p_r|H0(R|H0) and p_r|H1(R|H1) are adoped as probabilistic transition machanism for an observation R. (a) Find the risk (the average of the cost) in terms of P(Hi), Cij and P_r|Hi(R|Hi). (b) Find the detection rule of using the Bayes criterion. (c) Find the decesion rule of using maximun a posteriori (MAP) criterion Problem 2 (20%) Consider a detection problem with two hypothesis: the null hypothesis H0 with probability P(H0) and the alternative hypothesis H1 with probability P(H1). Let Cij be unknown, i=0,1. The conditional PDFs p_r|H0(R|H0) and p_r|H1(R|H1) are adoped as probabilistic transition for an observation R. (a) Find the detection rule of using Newman-Pearson criterion. (b) For using the detection rule of (a), how do you decide the required threshold value under the false-alarm probability <= α. Problem 3 (20%) In this problem, we consider that a single datum r has a normal distribution with mean and variance equal to a_j and σ^2. Under each M hypothesis Hj, for j= 0,1,...,M. Assume for simplicity that these mean value are arranged in ascending order ,a_1 < a_2 < ... < a_M. Moreover, let the hypothesis H0 denote that r has the normal distribution with mean and variance equal to zero and σ^2. Based on a signal observation of the random variable r, we try to decide among these M hypothesis with minimun probability error. (a) Find the optimun detector according to the minimization of total probability of error of the M+1 hypotheses are equally likely. (b) Find the decision region on the real line based on (a) Problem 4 (20%) Consider the estimation of a random variable "a". We perform a random experiment and take the independent observations r_i = a + n_i, i=1,2,...,M. Assume that n_i, i=1,2,...,M, are independent Gaussian noise samples with known variance σ^2 and zero mean, The random variable "a" is a Gaussian with mean m1 ≠0 and variance (σ_a)^2 (a) Find the optimum estimate "a" according to the mean-square error criterion (10%) (b) Find the optimum estimate of "a" according to the MAP criterion (5%) (c) Find the optimum estimate of "a" according to the maximum likelyhood(ML) criterion (5%) Problem 5 (20%) In this problem, we consider that an observation r=m+n, where m and n are two Poisson random variables with PDF given by N Pn(N)= λ exp(-λ) / N! with a known λ, N=0,1,2,.... , and M Pm|a(M|A) = A exp(-A) / (M!), M=0,1,2,... and a>=0 (a) Find the conditional PDF Pr|a(R|A) with R=0,1,2,... (7%) (b) Find the ML estimate of "a" (8%) (c) Is this ML estimate of "a" unbiased ? (5%) --

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