# Homework question: Conditional expectation on random variable itself

#### fireofsky

##### New Member
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

#### BGM

##### TS Contributor
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

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.