Does a continuous random variable always have a cdf? A pdf

probabilityprobability theoryrandom variables

I'm looking for the definition of a continuous random variable. Is it a r.v that has a continuous CDF? What I want to know is: is a continuous r.v defined via the CDF or PDF? i.e does there exist continuous r.v without a CDF or without a PDF?

Thanks!

Best Answer

A random variable is said to be continuous if its probability distribution function (AKA cumulative distribution function) is continuous. Notice the strangeness of this definition - we say a function is continuous if some other function is continuous!

Regarding your other questions:

  • CDFs are the more fundamental object. Densities don't always exist.
  • There are no random variables without a CDF. Random variables (measurable functions) induce (probability) measures, and the CDF is defined in terms of this measure.
  • There is a continuous random variable without a density - you just need to make the CDF weird. As mentioned, see the Cantor distribution.
Related Question