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# hw7 - EE 278 Statistical Signal Processing Homework#7 Due...

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EE 278 November 11, 2009 Statistical Signal Processing Handout #17 Homework #7 Due: Wednesday, November 18 1. Convergence examples. Consider the following sequences of random variables defined on the probability space (Ω , F , P), where Ω = { 0 , 1 , . . ., m - 1 } , F is the collection of all subsets of Ω, and P is the uniform distribution over Ω. X n ( ω ) = braceleftBigg 1 n ω = n mod m 0 otherwise Y n ( ω ) = braceleftBigg 2 n ω = 1 0 otherwise Z n ( ω ) = braceleftBigg 1 ω = 1 0 otherwise Which of these sequences converges to zero (a) with probability one, (b) in mean square, and/or (c) in probability? Justify your answers. 2. Convergencewithprobability1. Let X 1 , X 2 , X 3 , . . . be i.i.d. random variables with X i Exp( λ ). Show that the sequence of random variables Y n = min { X 1 , . . . , X n } converges with probabil- ity 1. What is the limit? 3. Convergence in probability. (Bonus) Let X 1 , X 2 , X 3 , . . . be a sequence of nonnegative random variables such that lim n →∞ E( X n ) = 0. a. Does the sequence X n converge in probability? If so, what is the limit? Justify your answer mathematically. b. Does the sequence Y n = 1 - e - X n converge in probability? If so, what is the limit? Justify your answer mathematically. 4. Vector CLT. The signal received over a wireless communication channel can be represented by two sums X 1 n = 1 n n summationdisplay j =1 Z j cos Θ j and X 2 n = 1 n n summationdisplay j =1 Z j sin Θ j , where Z 1 , Z 2 , Z 3 , . . . are i.i.d. with mean μ and variance

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