ch1-2A Covariance of random variables.xlsx - Covariance of two random variables cov(x y = E(x \u2212 \u03bcx(y \u2212 \u03bcy By definition Formula E(x \u2212 \u03bcx(y \u2212

# ch1-2A Covariance of random variables.xlsx - Covariance of...

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Covariance of two random variables By definition: cov(x, y) = Formula Simplified formula E(xy) = Proof = = The following example shows how to calculate the covariance of random variables x and y. Joint probability distribution of x and y y x 0 1 2 f(x) 0 0.14 0.06 0.10 0.30 1 0.07 0.12 0.25 0.44 2 0.05 0.09 0.12 0.26 f(y) 0.26 0.27 0.47 1.00 Marginal probability x f(x) y f(y) 0 0.30 0 0.26 1 0.44 1 0.27 2 0.26 2 0.47 E(x) = 0.96 E(y) = 1.21 var(x) = 0.5584 var(y) = 0.6859 Joint probability x y f(x,y) f(x,y) 0 0 f(0,0) 0.14 0 1 f(0,1) 0.06 0 2 f(0,2) 0.10 1 0 f(1,0) 0.07 1 1 f(1,1) 0.12 1 2 f(1,2) 0.25 2 0 f(2,0) 0.05 2 1 f(2,1) 0.09 2 2 f(2,2) 0.12 E[(x − μ x ) (y − μ y ) ] E[(x − μ x ) (y − μ y ) ] = ∑∑(x − μ x ) (y − μ y )f (x,y ) E[(x − μ x ) (y − μ y ) ] = E(xy) − μ x μ y ∑∑xyf(x,y ) E[(x − μ x ) (y − μ y ) ] = E(xy − x μ y − y μ x + μ x μ y ) E(xy) − μ y E(x) − μ x E(y) + μ x μ y E(xy) − 2 μ x μ y + μ x μ y E[(x − μ x ) (y − μ y ) ] = E(xy) − μ x μ y Joint probability xA yA Fxy
cov(x,y) = = cov(x,y) = 0.1184 =SUMPRODUCT((xA-Ex),(yA-Ey),Fxy) cov(x,y) = cov(x,y) = 0.1184 =SUMPRODUCT(xA*yA*Fxy)-Ex*Ey Conditional probability f(x|y) = f((x, y)/f(y) joint probability/marginal prob of y f(y|x) = f((x, y)/f(x)

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