For the first homework, there was a tricky question in problem 29, chapter 5. Here is the maximum of n random variables i.i.d. uniformly distributed on the unit interval . I gave a hint last week about the cumulative distribution function for the maximum, i.e.
is equal to the probability that all are smaller than ,
Then, we use independent to obtain that this probability is a product, of equal quantities since all random variables are identically distributed, i.e.
Then, the exercise ask the following
i.e. find a standardized version of that the maximum so that the cumulated distribution of that standardized version has a (non degenerated) limiting value. A hint is given in the answers, at the end of the book,
Actually, the question is not that simple (see here for the history of that question).
What I said during the course is that if is a random variable with finite variance, then
is a standardized (or normalized) version of , in the sense that it is centered, i.e.
and with a unit variance, i.e.
This is the kind of standardization (or normalization) that is used in the central limit theorem i.e. it is interesting when we study the core of our distribution (i.e. the mean).
Here we focus on the maxima (not on the expected value). Note that here
(up to some typing mistakes). Thus, our previous standardization would be
that can be simplified as
Hence, that random variable can be approximated by
since as . Here, it is then possible to get
since if , then (see the prof of the central limit theorem we got a few days ago).
But this is usually not the way we work with maxima. Actually, Fréchet, Fisher, Tippett, Gnedenko proved that the appropriate standardization to work with maxima is to consider
where is the cumulative distribution of the ‘s (the random variables used to build up the maximum). This work since the ‘s have a finite support, i.e. the are bounded, with an upper limit (here 1).
assuming that the density associated with exists. Hence, here the standardization becomes
which is exactly the one that John Rice is suggesting… And the proper motivation comes from extreme value theory, but it is a bit far away from what we shall see in that course…