What is the conditional probabilities and conditional expectation of $f(X)\vert X$

Question

There is only one random variable, $X$, and I want to find out the conditional probability and conditional expectation of $f(X)\vert X$.

Here is my thought:

Suppose $X$ is discrete, then $p(f(x)\vert x=x_{0})=\frac{p(f(x),x=x_{0})}{p(x=x_{0})}= \begin{cases} 1& \text{ if } f(x)= f(x_{0})\\ 0& \text{ if } f(x)\ne f(x_{0}) \end{cases} $

And the conditional expectation $E(f(x)\vert x=x_{0})=f(x_{0})$

Am I right and what if $X$ is contionuous?

Answer 1

Actually we have some basic properties of $E\{g(X)\vert Y=y \}$:

$E\{g_1(X)+g_2(X)\vert Y=y \}=E\{g_1(X)\vert Y=y \}+E\{g_2(X)\vert Y=y \}$

$E\{g_1(X)*g_2(Y)\vert Y=y \}=g_2(y)E\{g_1(X)\vert Y=y \}$