Question

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).

          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).

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).

Added by Victor P.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

01/16/2023

Step 1

We'll be working with the variance and covariance of two random variables, X and Y. Var(X + a) = VarX. Var(aX) = a VarX Var] = EX? (EX)? VarX = E(X(X - 1)) - (EX)(EX - 1). Var(X + Y) = VarX + VarY + 2Cov(XY): Cov(XY) = E(XY) _ (EX)(EY ): 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).