# Why copula based on CDF instead of PDF

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.

So,

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?

• Sklar's theorem guarantees the existence of a copula C for a joint CDF and marginals CDFs. – user289381 Jul 19 at 10:08
• @ping Thanks so much for your comment. So, why cdf is used instead of pdf. Is that because cdf is unique? – Maryam Jul 19 at 10:13
• 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. – user289381 Jul 19 at 10:17
• 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. – g g Jul 19 at 10:22
• @ping Thanks again. But why cdf is good in this case? what is the problem if we use pdf instead? – Maryam Jul 19 at 10:24

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 $$C$$.