Question

Prove the following properties using the definition of the variance and the covariance:

Q1. Operations with expectation and covariances Recall that the variance of randon variable X is defined as Var(X) Ξ E [X-E(X))2], the covariance is Cov(X, ) EX E(X))Y EY) As a hint, we can prove Cov(aX + b, cY)-ac Cov(X, Y) by ac EX -E(X)HY -E(Y)ac Cov(X, Y) In a similar manner, prove the following properties using the definition of the variance and the covariance: (a) Var(X)-Cov(X, X). (0.5 pt) (b) Cov(X, a) (0.5 pt) (c) Cov(aX, Y)=aCor(X,Y) (0.5 pt) (d) Cov(aX,bY)=abCor(X,Y). (0.5 pt) (e) Var(aX)a2Var(X). (0.5 pt)

Answer 1

a)
but a = 1 , c = 1 , b = 0
hence

Cov(X,X) = E((X - E(X))(Y -E(Y))
= Var(X)

b)
since a is constant
E(a) = a
hence
a - E(a) = 0
Cov(X,a) = 0

d)
replace b = 0 , c = b
hence
Cov(aX,bY) = ab Cov(X,Y)

e)
use a) and d)
to show
Var(aX) = a^2Cov(X,X)

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