I am trying to understand how the conditional probability is defined.
Along the way, I read the following statement from [page 145, Sec4.3]
: let $X$ and $Y$ be random variables.
We now consider this question in the case in which $X$ and $Y$ have a joint density $f$.
It made me think that two random variables whose joint density function does not exist. As far as I know, the probability density always exists by the Radon-Nikodym theorem.
So here are two questions:
- Does the probability density always exist by Radon-Nikodym theorem?
- If not, can I have an easy example of two random variables does not have its joint pdf?
Any comments/answers will very be appreciated.
For those who may argue that the discrete rv does not have its pdf, the pdf of a discrete rv can be defined via the Dirac-delta function.
Best Answer
Let $U$ be uniform on $[0,1]$; it has a density. Now let $V=U$. The pair $(U,V)$ is supported on the diagonal of the unit square, and does not have a joint density function.
Your instinct to use the Radon-Nikdym theorem is good, but that theorem has hypotheses that need checking. In this case the joint probability measure of the diagonal is $1$ but its 2-dimensional Lebesgue measure is $0$. The joint measure is not absolutely continuous w.r.t. Lebesgue measure, so the R-N plan cannot work.