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[ θalog M X ( θ )] d) bserve that the bound in (c) is true for ANY θ ≥ . Thus, conclude that Pr ( X > a ) ≤ exp(max θ ≥ [ θalog M X ( θ )]) (e) Compute the bound for the example in Problem (3) and comment....
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 Spring '07
 BARD
 Probability theory, Gaussian random variables, Random Processes Instructor

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