sta257week5notes - Theorem For g: R R E [g ( X )] = g (x )...

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week 5 1 Theorem For g : R Æ R If X is a discrete random variable then If X is a continuous random variable Proof: We proof it for the discrete case. Let Y = g ( X ) then ( ) [ ] ( ) ( ) = x X x p x g X g E () [] () () = dx x f x g X g E X
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week 5 2 Example to illustrate steps in proof • Suppose i.e. and the possible values of X are so the possible values of Y are then, 3 , 2 , 1 : ± ± ± X 2 X Y = ( ) 2 x x g = 9 , 4 , 1 : Y
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week 5 3 Examples 1. Suppose X ~ Uniform(0, 1). Let then, 2. Suppose X ~ Poisson( λ ). Let , then 2 X Y = X e Y =
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week 5 4 Properties of Expectation For X , Y random variables and constants, E ( aX + b ) = aE ( X ) + b Proof: Continuous case E ( aX + bY ) = aE ( X ) + bE ( Y ) Proof to come… If X is a non-negative random variable, then E ( X ) = 0 if and only if X = 0 with probability 1. If X is a non-negative random variable, then E ( X ) 0 E ( a ) = a R b a ,
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week 5 5 Moments •T h e k th moment of a distribution is E ( X k ). We are usually interested in 1 st and 2 nd moments (sometimes in 3 rd and 4 th ) Some second moments: 1. Suppose X ~ Uniform(0, 1), then 2. Suppose X ~ Geometric( p ), then () 3 1 2 = X E = = = 1 1 2 2 x x pq x X E
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week 5 6 Variance The expected value of a random variable E ( X ) is a measure of the “center” of a distribution. •T h e variance is a measure of how closely concentrated to center (μ) the probability is. It is also called 2nd central moment. Definition The variance of a random variable X is Claim : Proof: We can use the above formula for convenience of calculation.
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This note was uploaded on 08/27/2008 for the course STA 257 taught by Professor Hadasmoshonov during the Summer '08 term at University of Toronto- Toronto.

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sta257week5notes - Theorem For g: R R E [g ( X )] = g (x )...

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