Expectation of A given A,B Random variables

Question

So I am self-teaching myself some stuff about random variables and expectations for a course that I am going to take in the upcoming semester, and I found some resources online for properties that I would like to develop a better understanding of. The one that I have been looking at for the last two days can be found here: https://www.math.arizona.edu/~tgk/464_07/cond_exp.pdf

In the following, X, Y, Z are random variables, and $\mathbb{E}[X]$ is the expected value of X

In A.3 it introduces the idea of condition expectations on more than one random variable namely of this form $$ \mathbb{E}[X|Y = y,Z = z] $$ That usually gets denoted $$ \mathbb{E}[X|Y, Z] $$ I have also found that $ \mathbb{E}[X|X] = X $ which intuitively makes sense. I was wondering if this property expanded to a case like this: $$ \mathbb{E}[X|X, Y] = X? $$

It would be very much appreciated for a resource where I can find more info or really any help at all.

I read the above statement as "The expected value of X given X is x and Y is y is x" Is this, at least as an intuitive explanation, a good understanding of conditioning on multiple variables?

Answer 1

The expected value of X given X is x and Y is y is x

That is correct and specifically describes this: $E[X|X=x,Y=y]=x$. The expression $E[X|X,Y]=X$ is slightly different because this time there is no constant value $x$ associated with the RV $X$, and the result of the expression is a RV in terms of $X$. Intuitively, the expression $E[X|X,...] = X$, can be read as follows:

"If we know $X$ and some other things, expected value of $X$ will be what we know as $X$".