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problem set1 - ELEC 531 STATISTICAL SIGNAL PROCESSING...

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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 ) = sin 2 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 p Y ( y ), the probability density function for Y . Problem 1.3 A crucial skill in developing simulations for stochastic systems is random variable generation . Most computers (and environments like MATLAB) have software
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