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?
To find the covariance of XY and XZ, we can use the following formula:
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 write this expectation as a product of three expectations:
E[(XY)(XZ)] = E[X]E[Y]E[X]E[Z]
Given that E[X] = 0 and E[Y] = E[Z], we can simplify this to:
E[(XY)(XZ)] = 0 * E[Y] * 0 * E[Z] = 0
Next, let's calculate E[XY]E[XZ]:
E[XY]E[XZ] = E[X]E[Y] * E[X]E[Z]
Since E[X] = 0, this simplifies to:
E[XY]E[XZ] = 0 * E[Y]E[Z] = 0
Now, we can substitute these values back into the covariance formula:
Cov(XY, XZ) = E[(XY)(XZ)] - E[XY]E[XZ]
= 0 - 0
= 0
Therefore, the covariance of XY and XZ is zero.
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.