I do understand the mathematic behind probability density function( PDF) and cumulative distribution function (CDF). My problem starts when I try to understand why copula relies on CDF and not on PDF. I searched and found this:

the probability density function can be hard to work with directly. What I found confused me again.


1- Why copula relies on CDF and not PDf?

2- Why is it hard to work with PDF directly?

3- Why always CDF is used in integral transformation function. That is, why we cannot transform the variables into their uniform distribution using PDF?

  • $\begingroup$ Sklar's theorem guarantees the existence of a copula C for a joint CDF and marginals CDFs. $\endgroup$
    – user289381
    Commented Jul 19, 2020 at 10:08
  • $\begingroup$ @ping Thanks so much for your comment. So, why cdf is used instead of pdf. Is that because cdf is unique? $\endgroup$
    – Maryam
    Commented Jul 19, 2020 at 10:13
  • $\begingroup$ Sklar's theorem not only guarantees the existence but it also gives a simple way to estimate the copula from the CDF en.wikipedia.org/wiki/… If all marginal CDFs are continuous, C is unique. $\endgroup$
    – user289381
    Commented Jul 19, 2020 at 10:17
  • 1
    $\begingroup$ What do you mean by "copula relies on CDF and not on PDF"? Not every distribution may have a pdf but beyond that those two ways of describing a distribution are mathematically equivalent. What description works better depends on the problem you would like to solve. The ubiquitous bivariate scatter plots of copulas are a good example of the pdf-view. $\endgroup$
    – g g
    Commented Jul 19, 2020 at 10:22
  • $\begingroup$ @ping Thanks again. But why cdf is good in this case? what is the problem if we use pdf instead? $\endgroup$
    – Maryam
    Commented Jul 19, 2020 at 10:24

1 Answer 1


Sklar's theorem guarantees the existence of a copula $C$, given a joint CDF and marginal CDFs.

The theorem gives an easy method to determine the copula $C$.

Why do we use CDF instead of PDF?
"Every random variable has a CDF. Not every random variable has a PDF"



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.