Does every continuous random variable have a pdf?
Is there any random variable which is neither discrete nor continuous?
Here, by continuous random variable I meant those random variables for which probablity of a singleton set is 0.
Does every continuous random variable have a pdf
probability theory
Best Answer
If $X$ is a random variable with a Cantor distribution (i.e. the uniform distribution on the Cantor set $\subset [0, 1]$) then $X$ is a continuous r.v. without a pdf.
Lebesgue's decomposition theorem describes how any probability measure on $\Bbb{R}$ can be broken up into three parts with well-defined properties: a discrete part, a "pdf" part, and a singular part (one that's neither discrete nor has a pdf). So there's actually lots and lots of examples of continuous random variables that don't have pdfs.