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HW5_Spring11 - ELEC210 Spring 2011 Homework 5 1 Let X = U n...

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1 ELEC210 Spring 2011 Homework 5 1. Let n X U = where n is a positive integer and U is a uniform random variable in the interval [-1,1]. Find the cdf and pdf of X . 2. The amplitude of a radio signal X is a Rayleigh random variable with pdf: ( ) 2 2 /2 2 , 0, 0 x X x f x e x α α α = > > Find the pdf of ( ) 3 Z X + = ( ( ) ( ) max 0, X X + = ). 3. The input X to a communication channel is “-1” or “1”, with respective probabilities 1/3 and 2/3. The output of the channel Y is equal to: the corresponding input X with probability 1 e p p ; - X with probability p ; 0 with probability e p . (a) Describe the underlying space S of this random experiment and show the mapping from S to XY S , the range of the pair ( X , Y ). (b) Find the probabilities for all values of ( X , Y ). (c) Find [ ] P X Y , [ ] 0 P Y = . 4. A modem transmits a two-dimensional signal ( X , Y ) given by: ( ) cos 2 / 8 X r π = Θ and ( ) sin 2 / 8 Y r π = Θ where Θ is a discrete uniform random variable in the set {0,1,2,…,7}. (a) Show the mapping from S to XY S , the range of the pair ( X , Y ).
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