MATH255FINALCOMPLILATION.

DO NOT USE THIS PAGE FOR SOLUTIONS. WRITE ONLY YOUR FINAL ANSWERS ON THIS PAGE. Problem 1. [10 pts] You roll a fair six-sided die, and then you flip a fair coin the number of times shown by the die. Let X denote the number of heads obtained. Find E[X] and var(X). Numerical answers are required. Show your work in detail. B/X)= 7/4 Va (X) = 77/48 Problem 2. [10 pts] We obtain an observation X=+Z of a parameter and wish to estimate using a Bayesian approach. Suppose that is a discrete random variable with a PMF 1/2 for 0 = -1 and Z is a N(0,1) (standard normal) measurement noise random variable independent of . Calculate the Least Mean Squares (LMS) and Maximum A-Posteriori Probability (MAP) estimates of as a function of the observation realization value x. Express the LMS estimate in terms of the function tanhy = e-e. Show your derivation in full detail. ナ LMB()= tanh(x) Problem 3. [10 pts] Consider the pair of hypotheses H: X= Z H1: X=S+Z 1, x70xo where S and Z are independent random variables with fz(2) = e3, z2 0; 10, z<0; Le-es, 8≥0; and fs(s) = 0, 8 <0. Write the Neyman-Pearson test in the box below. Simplify the test as much as possible before writing. Compute the best possible type-II error probability ẞ for a given type-I error probability of a = 1/2. (The result for ẞ should be in a form that can be calculated readily by a calculator.) Neyman-Pearson test Hi X > M < Ho )2h-1(+- Solution of Problem 1 only. Let N be the outcome on the die. Then X= X₁'+ X₂t + XN where Xi= 1 if the itn toss is Heads, O otherwise 2 E[x] = E [£[XIN]] = E[N-E[X'1[ 71= 7 - E[N] E[x,] Var(X)= E [Var (XIN)] + Var (E(XIN)) where Var(XIN) = N. Var(x,) = N. (1) (2) E (XIN) = NE[X,] = N. Vac(x) = E[N+)+ Var(N) (01 L. Var (N( 2 4 Var(N) = 35 (discret--uniform PMF) (s). /2 Combining (1)-(4), we obtain Var(x) = 7 ++.35=£7 8 4 12 48 Solution of Problem 2 only. P@x (olx) = P@(o). fxI@ (20) Poto)fxi@ (o1o) + Po(1) fxI@(7|0) exp(-(x-0)"/2) = exp(-(x+1)/2) + xp.(-(x-1)/2) EX -(x-1)/2 -(x+1)/2 1 e e 1. e -(x+1)/2 -(x-11% te