In "Mathematical Foundations of the Calculus of Probability" by Jacques Neveu, the author says the following about distribution functions:

These functions, which are in fact of very little practical use (except in certain questions where the order structure of the real line plays a predominant role), should have disappeared a long time ago to the benefit of the ensemble definition of the notion of probability.

This sentiment (that they should have disappeared long ago) is also expressed by Erhan Ҫinlar in "Probability and Stochastics".

I have two closely related questions:

  • What is the justification for this view? I don't really see how you would come to this conclusion. How are distribution functions impractical?
  • Is this a commonly held sentiment among experienced mathematicians?

For the record: The distribution function of a RV $X$ is the function $x\mapsto \mu(-\infty\,..x]$, where $\mu$ is the distribution of $X$.

  • 1
    $\begingroup$ What does the author mean by the ensemble definition? $\endgroup$
    – littleO
    Feb 3 '17 at 19:32
  • $\begingroup$ @littleO The term "ensemble definition" seems to occur only this one time in the book. Judging from the material before this quotation, I think he simply means "definition of probability" (as in "normed finite measure") since he procedes to show, that there is bijection between probability measures on $(\mathbb R, \mathcal B(\mathbb R))$ and distribution functions (at this point random variables have not been introduced yet). $\endgroup$ Feb 3 '17 at 19:43
  • $\begingroup$ I wonder how this person feels about PDFs? If distributions and PDFs disappeared then most people would never learn probability. $\endgroup$
    – Michael
    Feb 4 '17 at 9:27
  • 1
    $\begingroup$ @Michael Maybe in a first course in probability, but not beyond that. The conventional definition of distribution is the pushforward measure $\mathbb P \circ X^{-1}$ of the the ambient probability measure $\mathbb P$ along $X$. $\endgroup$ Feb 5 '17 at 8:21
  • 1
    $\begingroup$ @StefanPerko Your guess is as good as mine, but I gather the reviewer means that convergence in distribution (which corresponds to pointwise convergence of the distributions functions) is barely touched on: this explains the reference to central limit theorems (typically, convergence in distribution) and characteristic functions (whose convergence pointwise is equivalent to convergence in distribution). $\endgroup$
    – Clement C.
    Feb 7 '17 at 15:35

I (think) the argument being made is how distribution functions only give you information about the probabilities of sets of the form $$X^{-1}( (-\infty, a])\,,$$ whereas the real source of interest is the values of the probability measure for sets of the form $$X^{-1}(A)$$ for any (Borel)-measurable set $A$.

I base this inference on the parenthetical comment from your quotation:

except in certain questions where the order structure of the real line plays a predominant role

In those certain cases I imagine that sets of the form $(-\infty, a]$ (and their inverse images under a random variable $X$) are more important than arbitrary Borel-measurable sets $A$.

This is, however, just a guess -- although I read Neveu's book a year or so ago, I don't quite remember what else (if anything) he mentioned about the topic. I have not yet read Cinlar's book, although I hope to do so in the future.

  • 3
    $\begingroup$ Mmh. But these intervals generate the Borel-$\sigma$-algebra and the distribution function uniquely determines the distribution (finite measure specified on generating $\pi$-system), so I don't really "get" it. $\endgroup$ Feb 3 '17 at 19:29

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