Derive the following elementary properties of expectation

Question

Assignment

Problem 1.

We showed in class that independence implies zero covariance. However, zero covariance does not imply independence. Consider random variables X and Y=X², where X is symmetric around zero. You can assume that X is a discrete RV. Symmetric around zero means Pr(X=x)=Pr(X=-x) for any x. Obviously, X and Y are not independent. Show that cov(Y, X)=0.

Problem 2.

Derive the following elementary properties of expectation, variance, and covariance (they will be repeatedly used in class).

1) The variance of a constant is zero: var(a) = 0.

2) The covariance is defined as cov(X,Y) = E[(X-μ_X) (Y - μ_Y)]. Show it can be expressed as cov(X,Y) = E(XY)- μ_Xμ_Y.

3) var(X+Y = var(X) + var(Y) + 2cov(X,Y).

4) cov(aX + bY, Z) = a cov(X,Z) + b cov(Y,Z), for constants a and b and random variables (X,Y,Z).

Problem 3.

For a random sample, (Xi ,Yi), i=1,...,n, show that (1/n) _i=1∑ⁿ (Y_i - Y‾) (X_i - X‾) is a consistent estimator of cov(Xi ,Yi). You may assume that relevant moments exist/finite for applications of LLN.