Question

Suppose that X, Y, and Z are independent random variables with unit variance. Furthermore, E[X] )= 0 and E[Y] = E[Z] = 2. Then, what's the covariance of XY and XZ?

Answer 1

To find the covariance of XY and XZ, we need to use the formula for covariance:

Cov(XY, XZ) = E[(XY)(XZ)] - E[XY]E[XZ]

First, let's calculate E[(XY)(XZ)]. Since X, Y, and Z are independent, we can multiply their expected values to get the expected value of the product:

E[(XY)(XZ)] = E[X^2 YZ]

Since X, Y, and Z have unit variance, E[X^2] = var(X) + E[X]^2 = 1 + 0^2 = 1. Therefore,

E[(XY)(XZ)] = E[X^2 YZ] = E[X^2]E[YZ] = 1 * E[YZ]

Next, let's calculate E[YZ]. Since Y and Z are independent, their expected value is equal to the product of their individual expected values:

E[YZ] = E[Y]E[Z]

Given that E[Y] = E[Z] = 2, we have:

E[YZ] = E[Y]E[Z] = 2 * 2 = 4

Now we can substitute these values into the covariance formula:

Cov(XY, XZ) = E[(XY)(XZ)] - E[XY]E[XZ]
= E[X^2 YZ] - E[XY]E[XZ]
= 1 * 4 - E[XY]E[XZ]

Since X, Y, and Z are independent, E[XY] = E[X]E[Y], and E[XZ] = E[X]E[Z]. Substituting these values:

Cov(XY, XZ) = 4 - E[X]E[Y]E[X]E[Z]
= 4 - 0 * 2 * 0 * 2
= 4

Therefore, the covariance of XY and XZ is 4.