# Ps2 - E[cos X 3 a quantity that is di±cult to compute analytically Problem 2 Two random variables X and Y have joint density p X,Y x,y = b Ke-2 x

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UCSB Fall 2009 ECE 235: Problem Set 2 Assigned: Monday October 5 Due: Monday October 12 Reading: Hajek, Chapter 1, Sections 3.1-3.4 Topics: Probability and random variables; jointly Gaussian random variables Reminder: Probability quiz in class on Tuesday, October 13 Agenda for this week: We will consolidate our probability review on Tuesday and then start on Chapter 3. We will start on Chapter 2 after covering part of Chapter 3. Practice problems (not to be turned in): The even numbered problems in Chapter 1, up to Problem 1.30. Compare your solutions with the solutions provided to make sure you understand the concepts. Problem 1: Consider the random variable X with CDF F X as in Problem 1.11. (a) Given a random variable U which is uniform over [0 , 1], specify in detail the transformation g ( U ) = F - 1 X ( U ) that maps it to a random variable with CDF F X . (b) Use simulations (e.g., using matlab) to estimate E [ X ]. Compare the result with the exact answer. (c) Once you are happy that your program works, use them to estimate
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Unformatted text preview: E [cos( X 3 )], a quantity that is di±cult to compute analytically. Problem 2: Two random variables X and Y have joint density p X,Y ( x,y ) = b Ke-2 x 2 + y 2 2 xy ≥ xy < (a) Find K . (b) Show that X and Y are each Gaussian random variables. (c) Express the probability P [ X 2 + X > 2] in terms of the Q function. (d) Are X and Y jointly Gaussian? (e) Are X and Y independent? (f) Are X and Y uncorrelated? (g) Find the conditional density p X | Y ( x | y ). Is it Gaussian? Problem 3: The random vector X = ( X 1 X 2 ) T is Gaussian with mean vector m = (-2 , 1) T and covariance matrix K given by K = p 9-2-2 4 P (a) Let Y 1 = X 1 + 2 X 2 , Y 2 =-X 1 + X 2 . Find cov ( Y 1 ,Y 2 ). (b) Write down the joint density of Y 1 and Y 2 . (c) Express the probability P [ Y 1 > 2 Y 2 + 1] in terms of the Q function. Problems 4-6: Problems 3.1, 3.3, 3.5...
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## This note was uploaded on 12/29/2011 for the course ECE 253 taught by Professor Brewer,f during the Fall '08 term at UCSB.

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