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 $f$ ?

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    $\begingroup$ The standard notation is $E[Y|X]$ and this random variable is defined as a measurable function of $X$ such that$$E[Y]=\int E[Y|x]\,\text{d}P(x)$$ $\endgroup$ – Xi'an Jun 9 '20 at 11:49
  • $\begingroup$ 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 $f(x)$ ? $\endgroup$ – Thomas Jun 9 '20 at 11:56
  • $\begingroup$ Not sure why the downvote anyway... $\endgroup$ – 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.

  • $\begingroup$ 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 ? $\endgroup$ – Thomas Jun 9 '20 at 10:12
  • $\begingroup$ @Thomas sorry, you're right, updated. $\endgroup$ – Tim Jun 9 '20 at 10:19
  • $\begingroup$ 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 ? $\endgroup$ – Thomas Jun 9 '20 at 12:01
  • $\begingroup$ Sorry but my question is all about notations, what is standard and what is not ... therefore I would like to be precise... $\endgroup$ – Thomas Jun 9 '20 at 12:04
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    $\begingroup$ @Thomas there is no strict rules with notations, just conventions as ones mentioned above and by Xi'an in the comment. You can always define what you mean by introducing a notation. $\endgroup$ – Tim Jun 9 '20 at 12:41

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