1. Prove the following properties of covariance using either the definition or the calculation form, i.e., Cov(X, Y) = E[(X - ?_X)(Y - ?_Y)] = E(XY) - E(X)E(Y). Note a, b, and c are non-random constants. (a) Cov(aX, bY) = ab Cov(X, Y) (b) Cov(X, bY + c) = b Cov(X, Y)
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Using the definition of covariance, we have: Cov(aX, bY) = E[(aX - E(aX))(bY - E(bY))] Now, we know that E(aX) = aE(X) and E(bY) = bE(Y), so we can substitute these values: Cov(aX, bY) = E[(aX - aE(X))(bY - bE(Y))] Now, we can distribute the constants a and Show more…
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Key Concepts
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Madhur L.
The Conditional Covariance Formula. The conditional covariance of $X$ and $Y$, given $Z$, is defined by $$ \operatorname{Cov}(X, Y \mid Z) \equiv E[(X-E[X \mid Z])(Y-E[Y \mid Z]) \mid Z] $$ (a) Show that $$ \operatorname{Cov}(X, Y \mid Z)=E[X Y \mid Z]-E[X \mid Z] E[Y \mid Z] $$ (b) Prove the conditional covariance formula $$ \operatorname{Cov}(X, Y)=E[\operatorname{Cov}(X, Y \mid Z)]+\operatorname{Cov}(E[X \mid Z], E[Y \mid Z]) $$ (c) Set $X=Y$ in part (b) and obtain the conditional variance formula.
Properties Of Expectation
Theoretical Exercises
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).
Sri K.
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