Homework question: Conditional expectation on random variable itself

#1
Hi:

I am confused by the following problem. (Please also find attachment)
Given X is a random variable, g is a real function of X mapped from R to R.
(g:X=>R:R)

(a) Find E[g(X)/X] (Find the expectation of E[g(X)/X] given X
(b) Find Var(g(X)/X]

Usually, we are only asked to find conditional expectation of g(X) given a different random variable such as Y.
For E[g(X)/X], I can only expand it as follow:
E[g(X)/X] =
∫g(x)P(g(x)/X=x)dx = ∫g(x) P(g(x),X=x)/P(X=x)dx

For Var(g(X)/X), it can be found using the general formula:
Var(X) = E(X^2)-(E[X])^2

Please help and give me some advice. Thanks!
 

BGM

TS Contributor
#2
Actually I should reply to this post earlier but just want to see if some one can provide a good explanation on this. I try my best :p

Based on what you have try seems you have learn the elementary, constructive definition of the conditional expectation. So here I just point out a simple constructive rule for this:

Let \( h(x) = E[g(X)|X = x] \) for all \( x \) within the support of \( X \). Then \( E[g(X)|X] = h(X) \)

So whenever you know how to evaluate \( E[g(X)|X = x] \) this, you know the functional form of \( E[g(X)|X] \), just replace it by the random variable.

And the main point of this question is, what is \( E[g(X)|X = x] \)?

In very layman speaking, once \( X = x \) is given, all \( X \) inside should be replaced by the deterministic constant \( x \). Or, more formally we should say \( g(X) \) is \( \sigma(X) \)-measurable. From this we can also see one fundamental property of conditional expectation:

\( E[g(X)|X] = g(X) \)

For the other question let you try again first.