Unformatted text preview: variables that are uniformly distributed between [0 , 1] . Using Chebyshev’s inequality and assuming x ≥ n 2 , a) Provide a bound on Pr (Σ n i =1 X i > x ) . b) Use the central limit theorem approximation to provide an approximation (in terms of the 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 6 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 ) and that, therefore, Pr ( X ≥ a ) ≤ e-[ θa-log M X ( θ )] ....
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This note was uploaded on 04/10/2011 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas.
- Spring '07