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 $X$.