# Expectation of A given A,B Random variables

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?

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 $$X$$".