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Example Class 5 Solution - THE UNIVERSITY OF HONG KONG...

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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 of X = E ( X ) Mean and variance Variance of X = Var ( X ) = ( ) 2 )) ( ( X E X E = E ( X 2 ) – E ( X ) 2 The mean is usually denoted by µ and the variance is usually denoted by σ 2 . 2. Moments and moment generating function 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 Moment ( ) r b X E ) ( . Let X be a random variable. The moment generating function of X is defined as Moment generating function M X ( t ) = E ( e tX ) if it exists. The domain of 𝑀 𝑋 ( 𝑡 ) are all real number t such that the expectation above is finite. And we have 𝑀 𝑋 ( 𝑟 ) (0) = 𝜕 𝑟 𝑀 𝑋 ( 𝑡 ) 𝜕𝑡 𝑟 𝑡=0 = E ( X r ) if M X ( t ) is differentiable at t = 0. Remark: moment generating function uniquely characterizes the distribution. 3. Some common continuous distributions Uniform distribution. Let X ~ U ( a , b ), where b > a . The probability density function (pdf) of X is
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