Suppose that X, Y, and Z are independent, with E[X]=E[Y]=E[Z]=2, and E[X^2]=E[Y^2]=E[Z^2]=5.
Find cov(XY,XZ).
To find the covariance between XY and XZ, we can start by using the definition of covariance:
cov(XY, XZ) = E[XY * XZ] - E[XY] * E[XZ]
Given that X, Y, and Z are independent, we know that the product XY and XZ will be equal to the product of their individual expectations:
E[XY * XZ] = E[X] * E[Y] * E[X] * E[Z] = (2 * 2) * (2 * 2) = 16
Next, we can substitute the given values into the equation to find the individual expectations:
E[XY] = E[X] * E[Y] = 2 * 2 = 4
E[XZ] = E[X] * E[Z] = 2 * 2 = 4
Finally, substituting the calculated values into the equation, we get:
cov(XY, XZ) = E[XY * XZ] - E[XY] * E[XZ] = 16 - (4 * 4) = 0
Therefore, the covariance between XY and XZ is 0.