Notation for Marginal Distribution Is there a satisfactory notation for marginal distributions? This would be one that clearly distinguishes the marginal distribution of $X$ relative to the joint distribution $(X,Y)$ from $X$ as it would be envisaged had $Y$ not been introduced.
One option might be $\mathbb{E}_Y(X|Y)$, but this seems to go beyond the basic definition of a marginal distribution.
 A: As pointed out by GeoMatt2, there is no need or even no meaning for a special notation.

"...clearly distinguishes the marginal distribution of $X$ relative to the
  joint distribution [of] $(X,Y)$ from $X$ as it would be envisaged had
  $Y$ not been introduced."

There is indeed some confusion in this question in that the marginal of $X$ is the marginal of $X$, no matter how it is derived:
$$p(x)=\int_\mathfrak{Y} p(x,y)\,\text{d}y=\int_\mathfrak{Y\times Z} p(x,y,z)\,\text{d}y\text{d}z=...$$for any completion mechanism one can dream of.
Introducing an extra-notation thus does not make sense when the function is the same. 
A: In my comments, I amplified on the comment of GeoMatt22.  The point was made by both of us that the marginal distribution for $X$ bore no relationship to $Y$ because $Y$ was integrated out.  Prior to Xi'an giving his answer, I suggested that I would like to summarize my comments in an answer because I did have a rational choice for a notation although it may not be standard.  Xi'an answered before the OP answered my comment. I am giving this answer because I am the only one so far to directly address what would be a rational notation for a marginal distribution for $X$. Here is a summary of the comments and the previous answer:

*

*GeoMatt22 rejected any notation that included $Y$ conditionally.  He made the important point that $Y$ is a nuisance variable and said that he just uses $p(x)$. He probably didn't make it a question because his choice is not standard.


*I then made three comments elaborating on GeoMatt22. Since my third comment included a notation that may not be standard but gets around the problem with $Y$ not being involved in the marginal for $X$.  Please look at the comments above before judging my answer.


*Xi'an enters an answer that acknowledges the key point made by GeoMatt22 and then expresses the point differently that a marginal integrates out the nuisance variable(s).  He does not suggest a notation because, as he says, it doesn't make sense.


*The OP accepts Xi'an's answer and gives a comment as to how he became confused but does not seem to have read my comments and possibly not GeoMatt22's either.
Now let me explain my answer by illustrating the ANOVA analogy. When you do an ANOVA, marginal sample averages are given a notation.  In the one-way ANOVA, the observations are denoted as $X_{ij}$ where $i$ represents the $i$th observation from the $j$th group, $i$ runs from 1 to $n_j$ for a given $j$ and $j$ from 1 to $p$ ($n_j$ observations in the group $j$ with $p$ groups).
So the notation $X_{i.}$ represents the average of the observations for fixed $i$, where the group effects are averaged out.
Likewise, $X_{.j}$ denotes for the $j$th group the average of the $X_{ij}$ where each of the $n$ observations for the $j$th group are averaged. So $X_{i.}$ = set of ($X_{ij}$) for fixed $i$, averaged over all $j$ from 1 to $p$ and $X_{.j}$ is the set of ($X_{ij}$) for fixed $j$ averaged over all $i$ in the $j$th group.
So by analogy for $f(x,y)$ denoting the joint density at $X=x$ and $Y=y$, let $f(x, .)$ represent the marginal density at $X=x$ with $Y$ integrated out and $f(.,y)$ the marginal density for $Y$ at $Y=y$ with $X$ integrated out.
