problem set1

Problem set1 - ELEC 531 STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 1 Due

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ELEC 531: STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 1 Due: September 4, 2007 Problem 1.1 Which of the following are probability density functions? Indicate your reasoning. For those that are valid, what is the mean and variance of the random variable? (a) p X ( x ) = e -| x | 2 (b) p X ( x ) = sin2 πx πx (c) p X ( x ) = ( 1 - | x | | x | ≤ 1 0 otherwise (d) p X ( x ) = ( 1 | x | ≤ 1 0 otherwise (e) p X ( x ) = 1 4 δ ( x + 1) + 1 2 δ ( x ) + 1 4 δ ( x - 1) (f) p X ( x ) = ( e - ( x - 1) x 1 0 otherwise Problem 1.2 A random variable X has cumulative distribution function P X ( x ) = [1 - exp( - 2 x )] u ( x ) , where u ( x ) is the unit-step function. (a) Calculate the following probabilities: Pr[ X 1] , Pr[ X = 2] , Pr[ X 2] . (b) Find p X ( x ), the probability density function of X . (c) Let Y be a random variable obtained from X as follows: Y = ( 0 X < 2 1 X 2 . Find
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This homework help was uploaded on 02/11/2008 for the course ELEC 531 taught by Professor Lexa during the Fall '07 term at Rice.

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Problem set1 - ELEC 531 STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 1 Due

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