Question

For each one of the following statements, indicate whether it is true or false.

Answer 1

Solution :

a). $\text{Var} (X|Y) =E ((X-E(X|Y))^2 |Y)$

  Now, if X = Y, then :

  $P(X|Y)=\left\{\begin{matrix} 1,& \text{if } x=y \\ 0, & \text{otherwise }\end{matrix}\right.$

Then, E[X|Y] = x = y

So, $\text{Var} (X|Y) =E((X-X)^2 |Y)$

                      $=E(0|Y)$

                      = 0

Therefore, this statement is TRUE.

b). If X = Y , then Var (X) = Var (Y)

And as Var (X|Y) = 0, so Var (X|Y) ≠ Var (X), except when all the elements of Y are same.

So this statement is FALSE.

c). As defined earlier,

  $\text{Var} (X|Y) =E ((X-E(X|Y))^2 |Y=y)$

  So, this statement is also TRUE.

d). The statement is TRUE because $\text{Var} (X|Y) =E ((X-E(X|Y))^2 |Y=y)$.

e). FALSE

   Because, $\text{Var} (X|Y) =E ((X-E(X|Y=y))^2 |Y=y)$