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

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. 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
.

10. Let X and Y be random variables. The conditional varance of Y given X, denoted Var(YIX), is defined as Show that Var(Y) = EVar(Y X) + Var(E[F(X). (This equality you are showing is known as the Law of Total Variance). Hint: Fron the Law of Total Expectation, you get Var(Y) =

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