# Name for special marginal distribution

Given $$P(X_1, \dots, X_N)$$ is there a name for the following two marginal distributions?

• The marginal, $$P(X_n)$$, including only the $$n$$-th variable
• The marginal, $$P(X_1, \dots, X_{n-1}, X_{n+1}, \dots, X_N)$$ including all but the $$n$$-th variable
• I was thinking of using "univariate marginal" for the first and "orthogonal projection" for the second marginal distribution. But maybe there is something more standard or more effective. Oct 21 '19 at 20:23

The first I would simply call the marginal distribution of $$X_n$$.
I don't think the second would usually be called a marginal distribution, since the name is generally used only for univariate distributions. It's true that you've marginalized out a variable, but it's still a joint distribution. Perhaps you could call it the "joint distribution excluding $$X_n$$" or something like that. I don't know of a standard name. I wouldn't recommend "orthogonal projection" since (to me at least) it's unclear what that means.
• Actually according to this a marginal should be any subset of the $N$ variables... Oct 26 '19 at 18:29