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hw7 - φ function for the expression in(a In other words...

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EE 351K Probability, Statistics, and Random Processes SPRING 2009 Instructor: Shakkottai/Vishwanath shakkott, [email protected] Homework 7 (Due 04/08/2009 on Wednesday) Problem 1 Let ( X 1 ,X 2 ) be jointly Gaussian random variables with ( X 1 ,X 2 ) N (2 , 1 , 9 , 16 , 0 . 3) . a) Determine the pdf of Y = X 1 + X 2 . b) Determine the joint pdf of ( W,Z ) where W = 3 X 1 + X 2 and Z = X 1 - X 2 . c) Determine an expression for E [ W | X 1 ] and E [ W | Z ] . Problem 2 Let ( X,Y ) be independent, identically distributed Gaussian random variables with mean 0 and variance σ 2 (i.e., X,Y N (0 2 ) ). a) Determine the pdf of R = X 2 + Y 2 . b) Determine the pdf of Θ = tan - 1 ( Y/X ) . (Hint: We have X = R cos(Θ) and Y = R sin(Θ) . ) Problem 3 Let X i ,i = 1 , 2 ,... be i.i.d. continuous random variables that are uniformly distributed between [0 , 1] . Using Chebyshev’s inequality and assuming x n 2 , a) Provide a bound on Pr n i =1 X i > x ) . b) Use the central limit theorem approximation to provide an approximation (in terms of the
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Unformatted text preview: φ function) for the expression in (a). In other words, provide an expression for Pr (Σ n i =1 X i > x ) using the central limit theorem. c) Compute (a) and (b) for n = 50 , and x = 80 and comment on your answers. Problem 4 Let X i ,i = 1 , 2 ,...,n be n i.i.d. random variables, with M x ( θ ) = E [ e θX ] . a) Show that for any θ ≥ , Pr ( e θX ≥ e θa ) ≤ E [ e θX ] e θa b) Argue that Pr ( X ≥ a ) = Pr ( e θX ≥ e θa ) . c) Using (a) and (b), show that Pr ( X > a ) ≤ e-[ θa-log M X ( θ )] d) bserve that the bound in (c) is true for ANY θ ≥ . Thus, conclude that Pr ( X > a ) ≤ exp(-max θ ≥ [ θa-log M X ( θ )]) (e) Compute the bound for the example in Problem (3) and comment....
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