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?




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 $X$.

  • $\begingroup$ thanks for the reply, but can we not see things like this ? that is, the joint distribution exists, but the probability mass on the straight with slope 1 in 2D space (that is, for (x0, x0) 2D vectors) are "infinity", and 0 elsewhere (that is, for (x0, x1) 2D vectors). then $f_{X,X}\left(x,x\right)$ could be defined as equal to $f_{X}\left(x\right)$ * the dirac function, and when integrating the integral of a dirac is 1...so it remains $f_{X}\left(x\right)$..and we have gotten our marginal $\endgroup$ – Romain227 Apr 19 '16 at 9:34
  • $\begingroup$ Yes, in a way we could write $dP_{(X,X)}(x,y)=f_X(x)dx\delta_x(dy)$ but be careful, this is not a function, there is no density. $\endgroup$ – Augustin Apr 19 '16 at 9:50

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