Moments - MOMENTS Mean of a Random Variable xk pk k E[ X= X...

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MOMENTS Mean of a Random Variable mean: [ ] kk k X x p discrete EX X x f (x)dx continuous −∞ = =
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Example Find the mean of a geometric random variable. pmf: 1 [ ] (1 ) , 1, 2, 3, . k k p PX k p p k = = = 1 1 1 ) k k kk X kp k p p p = = = −= ∑∑
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Example Find the mean of a uniform random variable. pdf: 1 ( ) with X f x axb ab ba = ≤≤ () 1 2 X b a X x f x dx x dx = = + =
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Example Find the mean of an exponential random variable. pdf: ( ) 0 with 0 as the arrival rate x X fx e x λ λλ = ≥> 0 () 1 X x X x f x dx x e dx = = =
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Higher Moments th moment: [ ] () m kk k mm m X xp m EX X x f x dx −∞ = = Expectation of a Function of a Random Variable [ ] () () k X g x p discrete EgX gX g x f x dx continuous −∞ ∆∆ Some useful techniques 1) [ ( )] [ ( )] for any constant E kg X kE g X = 2) [ ] [ ] [ ] EhX += +
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Second moment: 2 22 2 [] kk k X xp EX X x f (x)dx −∞ = = Variance ( ) ( ) 2 2 2 2 () ( ) k X X x Xp VAR X X X x X f x dx σ −∞ ≡− = Notice an easier way to find the variance: ( ) 2 2 X XX X X = −= X is called the standard deviation
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Coefficient of variation X X C X σ = For an exponential distribution, 1.
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Moments - MOMENTS Mean of a Random Variable xk pk k E[ X= X...

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