Let XY be two random variables. Using the definitions of variance and covariance, show that for any constants a and b: Var(aX + bY) = a^2Var(X) + 2abCov(X,Y) + b^2Var(Y) and Var(X - Y) = Var(X) - 2Cov(X,Y) + Var(Y)
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First, let's recall the definitions of variance and covariance: Variance: Var(X) = E[(X - E[X])^2] Covariance: Cov(X, Y) = E[(X - E[X])(Y - E[Y])] Now, let's find the variance of aX + bY: Var(aX + bY) = E[((aX + bY) - E[aX + bY])^2] We know that E[aX + bY] = Show more…
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1. (Variance and covariance) Let X and Y be two random variables. Prove the following properties of the variance and covariance: a) For any constant a, Var(X + a) = Var X, Var(aX) = a^2Var X. b) Var X = EX^2 - (EX)^2, c) Var X = E(X(X - 1)) - (EX)(EX - 1). d) Var(X + Y) = Var X + Var Y + 2Cov(X, Y). e) Cov(X, Y) = E(XY) - (EX)(EY).
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