6 Example Class 5

6 Example Class 5 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 5 Review 1. Expectation Assume that E ( X ) and E ( g ( X )) exist in the following, then E ( X ) E ( g ( X )) X is a discrete r.v. ) ( ) ( X x x xp ) ( ) ( ) ( X x x p x g X is a continuous r.v. ) ( ) ( X dx x xf ) ( ) ( ) ( X dx x f x g Mean and variance Mean of X = E ( X ) Variance of X = Var ( X ) = ( ) 2 )) ( ( X E X E- = E ( X 2 ) E ( X ) 2 The mean is usually denoted by and the variance is usually denoted by 2 . 2. Moments and moment generating function Moment Let r be a positive integer. The r th moment of X is E ( X r ). The r th moment of X about b is ( ) r b X E ) (- . Moment generating function Let X be a random variable. The moment generating function of X is defined as M X ( t ) = E ( e tX ) if it exists. The domain of g G is all real number t such that the expectation above is finite. And we have g G u = E ( X r ) if M X ( t ) is differentiable r times at t = 0. Remark: moment generating function uniquely characterizes the distribution....
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This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.

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6 Example Class 5 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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