Question: let x and y be random variables the conditional...

. Let X and Y be random variables. The conditional variance of Y given X, denoted Var(Y | X),
is defined as
Var(Y | X) = E[Y
2
| X] − E[Y | X]
2
.
Show that Var(Y ) = E[Var(Y | X)] + Var(E[Y | X]). (This equality you are showing is known
as the Law of Total Variance). Hint: From the Law of Total Expectation, you get Var(Y ) =
E[Y
2
] − E[Y ]
2 = E
h
E[Y
2
| X]
i
− E
h
E[Y | X]
i2
.