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homework6 - COMER ECE 600 Homework 6 1 Show that if X is a...

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C OMER ECE 600 Homework 6 1. Show that if X is a Cauchy random variable with parameter ° , then ω α ω - = Φ e ) ( X where ± X ( ² ) = E[e j ² x ] is the characteristic function of X. 2. Let X be a random variable with characteristic function ± X ( ² ). Show that | ± X ( ² )| takes on its maximum value at ² = 0. 3. The number of bytes N in a message has a geometric distribution with parameter p , i.e., ,... 3 , 2 , 1 , 0 , ) 1 ( ) ( p N = - = k p p k k where 0 ³ p ³ 1. (Note that this is an alternate version of the geometric pmf, and is slightly different from that presented in class). Suppose that messages are broken into packets of length M bytes. Let Q be the number of full packets in a message and let R be the number of bytes left over. Find the joint pmf and the marginal pmf’s of Q and R. Are Q and R independent? 4. Let X and Y be continuous random variables and φ φ φ φ cos Y sin X W sin Y cos X Z + - = + = where the angle φ is non-random. Find f ZW in terms of f XY . 5. Let R and ´ be independent random variables such that R has a Rayleigh density ) u( ) ( f 2 / 2 2 R 2 r e r r r σ σ - = and ´ is uniformly distributed on [- µ , µ
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