Problem 8.1 (Video 6.1, 6.2, Lecture Problem) Consider the following binary hypothesis testing scenario. (Note that all required integrals can be solved by calculating the areas of rectangles and triangles, so we are expecting exact answers.) 1 4 1 2 3 4 1 1 3 4 1 2 1 4 1 2 1 1 4 1 2 3 4 1 5 4 5 4 1 3 4 1 2 1 4 1 2 1 f Y|H1 (y) f Y|H0 (y) y f Y|H1 (y) f Y|H0 (y) y The hypothesis probabilities are P[H0] = 2/3 and P[H1] = 1/3. (a) Determine the ML rule. (b) Determine the MAP rule. (c) Determine the probability of error under the ML rule. (d) Determine the probability of error under the MAP rule. Problem 8.2 (Video 6.1, 6.2, 6.3, Quick Calculations) For each of the scenarios below, determine the requested quantities. (a) Under H0, Y is Gaussian(1, 1). Under H1, Y is Gaussian(+1, 1). Let P[H0] = 1/3 and P[H1] = 2/3. Determine the ML and MAP decision rules. (b) Under H0, Y is Exponential(1). Under H1, Y is Exponential(2). Let P[H0] = 1/2 and P[H1] = 1/2. Determine the likelihood ratio, the ML rule, and the probability of error under the ML rule. (c) Under H0, Y is Binomial(4, 1/2). Under H1, Y is Binomial(3, 1/2). Let P[H0] = 2/3 and P[H1] = 1/3. Determine the probability of error under the ML and MAP decision rules. 1 Problem 8.3 (Video 7.1, 7.2, Lecture Problem) Consider the following joint PDF fX,Y (x, y) = 4 x 0, y 0, x2 + y 2 1 0 otherwise. Note this is a uniform distribution over a quarter disk of radius 1. (a) Determine the MMSE estimator xMMSE(y) of X given Y = y. (b) Determine the Mean Square Error of the MMSE estimator E X xMMSE(Y ) 2 . Please compute the integrals in order to get a numerical answer. (c) Compute the LLSE estimate xLLSE(y) of X given Y = y. Please compute the integrals to get numerical answers. (d) Compute the Mean Square Error of the LLSE estimator. Please compute integrals to get numerical answers. Problem 8.4 Let X, Y be joint Gaussian random variables, with zero mean, and Var[X] = Var[Y ] = 2, Cov[X, Y ] = 1. (a) Determine the MMSE estimator xMMSE(y) of X given Y = y. (b) Let U, V be independent continuous random variables with U Uniform(-1,1), V Uniform(-1,1). Let Z = U 3 + V . compute linear least-squares estimate of Z based on observing U = u, denoted as Z LLSE(u). (c) Compute the Mean Square Error of both the MMSE estimate of Z given U and the LLS