Suppose that X , Y , and Z are independent, with E[X]=E[Y]=E[Z]=2 , and E[X2]=E[Y2]=E[Z2]=5 .
Find cov(XY,XZ) .
cov(XY,XZ)=
To find the covariance between XY and XZ, we need to use the formula for covariance:
cov(XY, XZ) = E[(XY - E[XY])(XZ - E[XZ])]
To simplify this equation, we need to calculate E[XY] and E[XZ].
Since X and Y are independent, E[XY] = E[X] * E[Y] = 2 * 2 = 4.
Similarly, since X and Z are independent, E[XZ] = E[X] * E[Z] = 2 * 2 = 4.
Now, we substitute these values into the covariance formula:
cov(XY, XZ) = E[(XY - 4)(XZ - 4)]
Since X, Y, and Z are independent, we can also say that XY and XZ are independent.
Therefore, E[(XY - 4)(XZ - 4)] = E[XY - 4XY - 4XZ + 16]
Now, we can distribute the expectation operator:
cov(XY, XZ) = E[XY] - 4E[XY] - 4E[XZ] + 16
Substituting the values we calculated earlier:
cov(XY, XZ) = 4 - 4(4) - 4(4) + 16
This simplifies to:
cov(XY, XZ) = -12
Therefore, the covariance between XY and XZ is -12.