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).
To find the covariance between XY and XZ, we can use the formula:
cov(XY, XZ) = E[XY*XZ] - E[XY] * E[XZ]
First, let's calculate E[XY]:
Since X and Y are independent, the expected value of their product is the product of their respective expected values:
E[XY] = E[X] * E[Y] = 2 * 2 = 4
Next, let's calculate E[XZ]:
Similarly, since X and Z are independent:
E[XZ] = E[X] * E[Z] = 2 * 2 = 4
Now, let's calculate E[XY * XZ]:
Since X, Y, and Z are independent, we can write:
E[XY * XZ] = E[X] * E[Y] * E[X] * E[Z] = 2 * 2 * 2 * 2 = 16
Finally, we can substitute these values into the covariance formula:
cov(XY, XZ) = E[XY * XZ] - E[XY] * E[XZ] = 16 - 4 * 4 = 16 - 16 = 0
Therefore, the covariance between XY and XZ is 0.
To find the covariance between XY and XZ, we can use the formula:
cov(XY, XZ) = E[(XY - E[XY])(XZ - E[XZ])]
First, let's find E[XY] and E[XZ]:
E[XY] = E[X] * E[Y] = 2 * 2 = 4
E[XZ] = E[X] * E[Z] = 2 * 2 = 4
Now let's find the covariance:
cov(XY, XZ) = E[(XY - 4)(XZ - 4)]
Since X, Y, and Z are independent, we can split the expectation into separate terms:
cov(XY, XZ) = E[XY * XZ - 4XZ - 4XY + 16]
Using the linearity of expectation, we can split it further:
cov(XY, XZ) = E[XY * XZ] - E[4XZ] - E[4XY] + E[16]
Since X, Y, and Z are independent, E[XY * XZ] = E[XY] * E[XZ] = 4 * 4 = 16.
E[4XZ] = 4 * E[X] * E[Z] = 4 * 2 * 2 = 16.
E[4XY] = 4 * E[X] * E[Y] = 4 * 2 * 2 = 16.
E[16] = 16.
Substituting these values back into the covariance formula:
cov(XY, XZ) = 16 - 16 - 16 + 16
Simplifying:
cov(XY, XZ) = 16 - 16 - 16 + 16 = 0
Therefore, the covariance between XY and XZ is 0.