Question

Q2. More about operations with expectation and covariances Recall that the variance of random variable X is defined as Var(X) Ξ E 1(X-E(X))2」, the covariance is Cor(X, Y-E (X-E(X))(Y-E(Y)), and the correlation is Corr(X,Y) Ξ (a) What is the value of EX-E(X))? (Hint: Let μ denote E(X). Then, the parameter μ is a unknown, but fixed value like a constant.) (0.5 pt) b) The following is the proof that Var(X) E(X2) E(X)2: -E(x)-E(x)2 In a similar way, prove that Cov(X, Y)-E(XY) - E(XEY). (0.5 pt) (c) Prove the following statement. (0.5 pt) "If 0 < Var(X) < oo and 0 < Var(Y) < oo, Corr(X, Y) = 0 implies Cov(X, Y)-0."

Answer 1

a)

E(X E(X))

E(X) is constant   and E(aX) = a E(X)

hence

E(X E(X)) = E(X)E(X) = E(X)^2

b)

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

$= E( X Y - Y E(X) - XE(Y) &plus; E(X)E(Y))$

$= E( X Y) - E(Y) E(X) - E(X)E(Y) &plus; E(X)E(Y)$

$= E( X Y) - E(X)E(Y)$

c)

since

Cov(X,Y)/ sqrt (Var(X)Var(Y)) = Cor(X,Y)

since

Cor(X,Y) = 0

hence

Cov(X,Y) = 0