Suppose that X, Y, and Z are independent random variables with unit variance. Furthermore, E[X] )= 0 and E[Y] = E[Z]. Then, what's the covariance of XY and XZ?

Answer 1

To find the covariance of XY and XZ, we can use the property that Covariance(aX, bY) = ab * Covariance(X, Y) for any constants a and b.

First, let's find the covariance of X and Y. Since X and Y are independent, Cov(X, Y) = 0. Thus, Cov(X, Y) = 0 * 1 = 0.

Next, let's find the covariance of X and Z. Since X and Z are also independent, Cov(X, Z) = 0.

Using the property mentioned above, we can find the covariance of XY and XZ:

Cov(XY, XZ) = Covariance(X, X) * Covariance(Y, Z) = Cov(X, X) * Cov(Y, Z) = Var(X) * 0 = 1 * 0 = 0.

Therefore, the covariance of XY and XZ is 0.