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20-Joint_Cov_Corr - IE 111 Spring Semester 2010 Properties...

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IE 111 Spring Semester 2010 Properties of Expected Value, Variance, and Covariance Previously in this class we have seen some properties of expected value and variance. We are now in a position to prove these properties, and develop some new properties for covariance. 1. for constants a and b, E(aX+b) = aE(X)+b E(aX+b) = Rx (ax+b)f X (x)dx = Rx axf X (x)dx + Rx bf X (x)dx = a Rx xf X (x)dx + b Rx f X (x)dx = aE(X) + b(1) 2. E(X+Y) = E(X) + E(Y) E(X+Y) = Rx Ry (x+y)f XY (x,y)dydx (by definition) = Rx Ry xf XY (x,y)dydx + Rx Ry yf XY (x,y)dydx (expand the sum) = Rx Ry xf XY (x,y)dydx + Ry Rx yf XY (x,y)dxdy (switch integration order on 2nd term) = Rx x Ry f XY (x,y)dydx + Ry y Rx f XY (x,y)dxdy = Rx xf X (x)dx + Ry yf Y (y)dy (by definition of getting marginal from joint) = E(X) + E(Y) 3. E(a 1 X 1 + a 2 X 2 + ... + a p X p + b) = a 1 E(X 1 )+ a 2 E(X 2 )+ ... + a p E(X p )+ b By the repeated application of 1. and 2. 4. V(X) = E [( X-E(X) ) 2 ] By definition of expectation and variance.
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