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CEE 3040 added notes Lec12

CEE 3040 added notes Lec12 - Derivation of the...

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CEE 3040 added notes Lec12 So-called “new example” Details for the W = X + Y distribution: note that although X and Y are uniform, W is non-uniform as you move across the square of possible values from (0,0) to (1,1), i.e., there is more area where you can have values 0.75 < w < 1 than values 0 < w < .25. Intuitively, the following pdf makes sense: f-w(w) = w for 0 < w < 1 (1-w) for 1 < w < 2 0 elsewhere Integrating f-w(w) by parts gives the result I showed in class. Demonstrating if X, Y independent then E[XY] = E[X]E[Y] By definition of expected value: dxdy f xy XY E xy ∫ ∫ = ] [ By definition of independence of X and Y, and rearranging integral: ] [ ] [ ] [ Y E X E dy yf dx xf dxdy f f xy XY E f f f y x y x y x xy = = = =
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Unformatted text preview: ∫ ∫ ∫ Derivation of the third property of covariance (p.4/6): ( 29 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Y Var b Y X abCov X Var a b ab a Y E b XY abE X E a b a Y b abXY X a E bY aX E bY aX E bY aX Var y y x x y x 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 , 2 2 2 2 ] [ + + =---+ + = +-+ + = +-+ = + μ Example of linear operations on Variance of binomial: Let X ~ Binomial Then: [ ] [ ] ∑ = i X Var X Var where each X i is distributed according to Bernouilli distribution [ ] ( 29 [ ] [ ] ( 29 p np X Var n X Var p p X Var i i i-= ⋅ =-= ∑ 1 1 Thus, the known variance of the binomial distribution np(1-p) results....
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