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.