# Distribution of extremal values

If an item follows normal distribution, average also follows normal distribution. What about minimum and maximum?

• You might want to look into this book. – mpiktas Apr 18 '11 at 6:57
• @user4211, do you ask about distribution of minimum and maximum of any sample distribution, or only normal? – mpiktas Apr 18 '11 at 6:58

You should have a look at the order statistics. Here is a very brief overview.

Let $X_{1}, \ldots X_{n}$ be an i.i.d. sample of size $n$ drawn from a population with distribution function $F$ and probability density function $f$. Define $Y_{1}=X_{(1)}, \ldots, Y_{r} = X_{(r)}, \ldots, Y_{n}=X_{(n)}$, where $X_{(r)}$ denotes the $r$th order statistic of the sample $X_{1}, \ldots X_{n}$, i.e., its $r$th smallest value.

It can be shown that the joint probability density function of $Y_{1}, \ldots, Y_{n}$ is

$f_{X_{(1)}, \ldots, X_{(n)}}(y_{1}, \ldots, y_{n}) = n! \prod_{i=1}^{n} f(y_{i})$ if $y_{1} < y_{2} < \ldots < y_{n}$ and $0$ otherwise.

By integrating the previous equation we get

$f_{X_{(r)}}(x) = \frac{n!}{(r - 1)! (n - r)!} f(x) (F(x))^{r-1} (1 - F(x))^{n - r}$

In particular, for the minimum and maximum, we respectively have

$f_{X_{(1)}}(x) = n f(x) (1 - F(x))^{n-1}$

$f_{X_{(n)}}(x) = n f(x) (F(x))^{n - 1}$

• +1, I've edited a small mistake in the second last formula. – mpiktas Apr 18 '11 at 6:59
• Thanks ocram, the answer is impressive so I checked as good answer but now can you make it in plain english thanks :) By the way how do you put equation in stackexchnage ? – user4211 Mar 5 '12 at 8:07
• What do you mean exactly? You asked for the pdf's of the minimum and of the maximum, and these two are given by $f_{X_{(1)}}$ and $f_{X_{(n)}}$, respectively. So, if you draw many many samples and compute the min for each, then you end up with a random variable with pdf $f_{X_{(1)}}$. Is it ok? – ocram Mar 6 '12 at 9:21

You might also want to read up on the generalized extreme value (GEV) distribution. It turns out that as $n\rightarrow\infty$, the (shifted and scaled) distribution of the maximal value of the sample converges to one of the three special cases of the GEV distribution.

• Great link will read it – user4211 Mar 5 '12 at 8:08

The sum of Gaussians is Gaussian. That is why the average is normal. The distribution of any non-linear function of (finitely many) Gaussians need not be Gaussian, and it usually isn't. Such is the case of the maximum function. To approximate the maximum of a multivariate Gaussian, Hothorn is a good place to start.

• very interesting will read hothorn – user4211 Mar 5 '12 at 8:08