TITLE: joint distribution of x with..itself
QUESTION [6 upvotes]: I have a weird question about probability and density functions : 
Let's take a random variable X whose p.d.f exists and let's denote it $f_{X}\left(x\right)$.
Does the definition of the joint probability $f_{X,X}\left(x,x\right)$ exist ?
clearly it's not continuous but i wanted to "check" that the marginal of X ($f_{X}\left(x\right)$) would be the integral of this joint distribution...
Can you give me more insight about it?
thanks,
Romain

REPLY [6 votes]: You can define the probability distribution of $(X,X)$ but there is no density with respect to the Lebesgue measure of $\mathbb{R}^2$, because $(X,X)$ is supported by $\Delta=\{(x,x),\, x\in\mathbb{R}^2\}$, whose Lebesgue measure in $\mathbb{R}^2$ is $0$. So the pdf of $(X,X)$ does not exist, it's a degenerate distribution.
However, you can calculate it's cdf, which always exists.
$$P(X\leq x,X\leq y)=P(X\leq\min(x,y))=F_X(\min(x,y))$$
where $F_X$ denotes the cdf of $X$.