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Unformatted text preview: ) = P (  X  1 ) = 1. P (  X  < ) = P (  X  < 1 ) = 0. Example 3 : Let a > 0, 0 < p < . Consider a discrete random variable X with p.m.f. P ( X = a ) = p , P ( X = 0 ) = 1 2 p , P ( X = a ) = p . Then = E ( X ) = 0, 2 = Var ( X ) = E ( X 2 ) = 2 p a 2 . Let k = p 2 1 > 1. Then k = a . & P (  X  k ) = P (  X  a ) = 2 p = 2 1 k . P (  X  < k ) = P (  X  < a ) = 1 2 p = 2 1 1 k. Jensens Inequality: If g is convex on an open interval I and X is a random variable whose support is contained in I and has finite expectation, then E [ g ( X ) ] g [ E ( X ) ]. If g is strictly convex then the inequality is strict, unless X is a constant random variable. & E ( X 2 ) [ E ( X ) ] 2 Var ( X ) 0 & E ( e t X ) e t E ( X ) & M X ( t ) e t & E [ ln X ] ln E ( X )...
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This note was uploaded on 10/22/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Fall '08 term at University of Illinois at Urbana–Champaign.
 Fall '08
 AlexeiStepanov
 Probability

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