The largest, or smallest, observation in a sample has one of three possible distributions. This is another example of convergence in distribution.

The *average of n* samples taken from *any* distribution with finite mean and
variance will have a *normal* distribution for large *n*. This is the
CLT. The
*largest* member of a sample of size *n* has a **LEV**,
**Type I largest extreme value** distribution, also called **Gumbel**, distribution, regardless of the parent
population, **IF** the parent has an unbounded tail that decreases at least as
fast as an exponential function, and has finite moments (as does the normal,
for example).

The LEV, has pdf given by

where and , are the location and scale* parameters, respectively, and > 0.

Similar sampling of the *smallest* member of a
sample of size n produces an **SEV**, **Type I**
**smallest extreme value**
distribution, with density

as *n* increases. There are two other extreme value distributions. If not all
moments exist for the initial distribution, the largest observation follows a
**Type II** or **Frechet** distribution. If the parent density has a bounded tail,
the smallest observation in a sample of size *n*, has a **Type III**, or
**Weibull**
distribution of minima, as *n* increases. Examples are smallest samples taken
from lognormal, Gamma, Beta or Weibull distributions.

The Weibull distribution is most easily described by its cdf:

where
is
the *scale* (not *location*) parameter, and
is
the shape* *parameter. Weibull is *not* a location, scale
density*.

Notice that if *x* has a Weibull distribution, then log_{e}(*x*) is SEV,
so SEV is to Weibull, as normal is to lognormal. Type I and Type III
limiting distributions are useful in describing physical phenomena where the
outcome is determined by the behavior of the best, or worst, in the sample.

***** A probability density is a **location, scale density** if it can take the form
of

where is a proper density and does not depend on any unknown parameters.
is the location parameter and
is the scale parameter. The
Normal, or Gaussian, is a special case with =
= mean, and =
=
standard deviation. Although location, scale densities are sometimes written
using
, as generic parameters,
*these do not*, in general, refer to the
mean and standard deviation of a location, scale density. Note again that
*not all densities are location, scale*.

A **proper density** is one for which
and
.