For a probability distribution, what is the most fundamental object : the probability density function or the cumulative distribution function ?
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4CDF. Not all distributions can be represented by a PDF. – Michael Dec 24 '21 at 20:15
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1The pdf may not exist but the cdf always exists . https://math.stackexchange.com/questions/98801/probability-distribution-function-that-does-not-have-a-density-function – Mr. Gandalf Sauron Dec 24 '21 at 20:15
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thanks a lot to the two of you for having shared these extremely useful informations. – Mathieu Krisztian Dec 24 '21 at 20:31
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This is mostly to exampnd on the great comments, which address this pretty well. More generally, if you get into a little measure theory, the CDF can be used to define the underlying probability*measure* on the range of the random variable. Specifically -- the CDF assigns numbers to intervals in a consistent way whereas the density (if it exists) does not (you need to integrate it).
Therefore, the CDF is an example of a measure $\mu$ on half-open intervals $(-\infty,x]$ (in the one dimensional case).
In contrast, the density requires that the measure is differentiable. A more general version of the usual calculus derivative is the Radon-Nikodym derivative, which handles cases where you are not restricted to the reals.