In the comments of this question, it was pointed out that, when comparing two distributions, it is more natural and more general use the cumulative distribution (CDF) instead of the distribution (PDF).

The question is, why? I.e. what are the advantages (and/or disadvantages) of using the CDF instead of the PDF that make it more "more natural and general"?

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    $\begingroup$ In a well-defined sense "most" distributions do not even have a PDF. In another well-defined sense "most" distributions that have a PDF do not have a unique PDF (they can differ on sets of measure zero). Thus this question will have interesting answers only for those who focus exclusively on distributions that have everywhere continuous PDFs, in which case there may be some room for debate. $\endgroup$ – whuber Feb 18 '14 at 8:57

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