What is the correct notation for "expected value of function, given that we know the variable"? Take a function $f(x)$, where $x$ is a random variable, but $f$ is also a "random function", meaning that even if we know $x$, we don't know $f$ with certainty. (I don't know if it is even acceptable to say this. If it isn't, then just think of $f(x,y)$ instead, where $y$ is also a random variable).
I want two particular things:


*

*The expected value of $f$, given that we know $x$ has some particular value, say 5.

*the expected value of $f$, as a function of $x$, which we don't know. 


How do we write these things down as a formula?
 A: You can consider $F$ to be a parametric random variable depending on the parameter $x$ and denoted $F_x$ or $F(x)$.
Its pdf would be
$$p_x(f):=\mathbb P(F_x=f).$$
Now if $x$ is a random variable, you can consider the conditional distribution
$$p_x(f):=\mathbb P(F_X=f|X=x)$$
versus the ordinary distribution
$$p_X(f):=\mathbb P(F_X=f).$$
Then
$$E(F_x)=\int f\,p_x(f)\,df\\\text{ vs. }\\E(F_X)=\int f\,p_X(f)\,df=\int f\,\mathbb P(F_X=f\land X=x)\,df\,dx$$
A: In quantum mechanics we would subscript the bra-ket notation
f(x,y) - function of two random variables
$e(x) = \langle f(x,y) \rangle_y = \int_{y \in \Omega} f(x,y)P(y)dy $ - expectation value with respect to variable $y$
$e(y) = \langle f(x,y) \rangle_x = \int_{x \in \Omega} f(x,y)P(x)dx $ - expectation value with respect to variable $x$
I have never heard of "random functions", you will have to be more precise with what you mean by it. If you mean that the function itself can be random, and you can't parameterize that randomness easily using a random variable parameter $y$, then the question is really hard
A: Instead of using the term "random function", you can just say that you are given a statistics $f(X,Y)$ of two random variables $X,Y$. Mathematically, a statistics is just a composite function of random variables.
For your first question, it is common to write $E_{Y}f(x,Y)$ (here "$x$" denotes a realization of $X$); the subscript is used to inform the reader with respect to which random variable we take expectation.
For the other one, just write $Ef(X,Y)$; the absence of subscripts says enough to the reader.
As an appendix, random functions are another type of mathematical objects. 
