1. Prove the following properties of covariance: • cov(X, Y) = cov(Y, X) • cov(X,X) = Var(X) (15 pts) • cov(aX, Y) = a cov(X, Y) for any constant a • cov(X+c, Y+d) = cov(X,Y) where c and d are constants
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Step 1: By definition, the covariance of two random variables X and Y is given by: $$cov(X,Y) = E[(X-E[X])(Y-E[Y])]$$ 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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The conditional covariance of X and Y, given Z, is defined by Cov(X, Y|Z) = E[(X - E[X|Z])(Y - E[Y|Z])|Z] Show that Cov(X, Y|Z) = E[XY|Z] - E[X|Z]E[Y|Z] Prove the conditional covariance formula Cov(X, Y) = E[Cov(X, Y|Z)] + Cov(E[X|Z], E[Y|Z]) Show that the variance of E[X|Z] cannot be larger than the variance of X.
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Prove the following properties of covariance using either the definition or the calculation form; i.e. Cov(X,Y) = E[(X - E(X))(Y - E(Y))]. Note a, b, and c are non-random constants. Cov(aX, bY) = ab Cov(X,Y) Cov(X, bY + c) = b Cov(X,Y)
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