# Notation about conditional expectation $E[Y|X]$

Given $$X,Y$$ real random variables, we know that $$E[Y|X]$$ is X measurable and that there is a Lebesgue measurable function $$f : \mathbf{R} \rightarrow \mathbf{R}$$ such that $$E[Y|X]=f(X)$$ almost everywhere.

Is there any explicit and standard notation for the function $$f$$ as a function of $$X,Y$$ ? Something like $$\mu_{Y|X}$$ (I am just inventing here), so that one could write $$\mu_{Y|X}(x_1)$$ instead of $$f(x_1)$$ and therefore make it explicit the meaning of $$f$$ ?

• The standard notation is $E[Y|X]$ and this random variable is defined as a measurable function of $X$ such that$$E[Y]=\int E[Y|x]\,\text{d}P(x)$$ – Xi'an Jun 9 '20 at 11:49
• This looks like the law of total expectation. Why are your reporting it ? Maybe you want to suggest that $E[Y|x]$ is the standard notation for my $f(x)$ ? – Thomas Jun 9 '20 at 11:56
• Not sure why the downvote anyway... – Thomas Jun 9 '20 at 13:13

Same as we can use $$f_X(x)$$ for the probability density $$f$$ of random variable $$X$$ evaluated at $$x$$, you can use notation like $$f_{Y|X}(y|x)$$ or $$\mu_{Y|X}(y|x)$$ to denote conditional things (e.g. densities, expected values etc.). See examples here or here.
• Great thanks! There is a leap on the notation though, right? Because $Y|X$ is not a well defined random variable. Has $\mu_{Y|X}(y)$ the same meaning of my question ? In this case the natural argument should be $x$ not $y$ or am I wrong ? And how is $f_{Y|X}(y)$ defined ? – Thomas Jun 9 '20 at 10:12
• Thanks Tim, still in my mind it should be $\mu_{Y|X}(x)$ and, if I understand correctly your notation, $f_{Y|X}(y|x)$ is a shorthand for a two-variable function $f_{Y|X}(y,x)$. Than $\mu_{Y|X}(x)$ should have the meaning of the mean of $Y$ along all events where $X=x$, so that depends only on one variable. Whereas (tell me if I am wrong) you are using $y|x$ as a shorthand for $(y,x)$ and $f_{Y|X}(y,x)$ is the probability that $Y=y$ given that $X=x$. We have than $\mu_{Y|X}(x)=\int dy \ y f_{Y|X}(y|x)$ . Is this what you are suggesting ? – Thomas Jun 9 '20 at 12:01