Proof of a property of conditional expectation

Question

I'm new here and this is my first question.

I'm struggling with a proof for my statistics course.

I need to show the following proposition:

$E[Y^2|\bar{X}=\bar{x}]\geq[E(Y|\bar{X}=\bar{x})]^2$

I tried to show it using Jensen's inequality but my teacher told me that is not correct.

Thank you :)

Answer 1

Hint:

• This can be shown using $\textrm{Var}(Y \mid \bar X = \bar x) \ge 0$

• So if you want more detail, expand and rearrange $\mathbb E[(Y - \mathbb E[Y \mid \bar X = \bar x])^2 \mid \bar X = \bar x] \ge 0$, which is true since it is the conditional expectation of a square term

Answer 2

Not to subvert your teacher, but there is a conditional form of Jensen's inquality, which in your situation would be: For a function $g$ defined and convex on the range of $Y$, $$ E[g(Y)\mid \bar X =\bar x]\ge g\left(E[Y\mid \bar X = \bar x]\right). $$ (Of course, $Y$ needs to have a finite expectation.)