The
probability distribution
with probability density
depending on a parameter
.
The
distribution function
of the Maxwell distribution has the form
where
is the standard
normal distribution
function. The Maxwell distribution has positive coefficient of skewness;
it is unimodal, the unique mode occurring at
.
The Maxwell distribution has finite moments of all orders;
the mathematical expectation and variance are equal to
and
,
respectively.
If
,
and
are independent random variables having the normal distribution with parameters
and
,
then the random variable
has a Maxwell distribution with density
(*).
In other words, a Maxwell
distribution can be obtained as the distribution of the length of
a random vector whose Cartesian coordinates in three-dimensional
space are independent and normally distributed with parameters
and
.
The Maxwell distribution with
coincides with the distribution of the square root of a variable having the
-distribution
with three degrees of freedom (see also
Rayleigh distribution).
The Maxwell distribution is widely known as the velocity distribution
of particles in statistical mechanics and physics. The distribution was first
defined by
J.C. Maxwell
(1859)
as the solution of the problem
on the distribution of velocities of molecules in an ideal gas.