A common name for a class of probability distributions, arising as
an extension of the idea of
"equally possible outcomes"
to the continuous case. As with the
normal distribution,
the uniform distribution appears in probability theory as an
exact distribution in some problems and as a limit in others.
The uniform distribution on an interval of the line (the rectangular distribution).
The uniform distribution on an interval
,
,
is the
probability distribution
with density
The concept of a uniform distribution on
corresponds to the representation of a random choice of a point from
the interval. The mathematical expectation and variance of
the uniform distribution are equal, respectively, to
and
.
The distribution function is
and the characteristic function is
A random variable with uniform distribution on
can be constructed from a sequence of independent random variables
taking the values 0 and 1 with probabilities
,
by putting
(
are the digits in the binary expansion of
).
The random number
has a uniform distribution in the interval
.
This fact has important statistical applications, see, for example,
Random and pseudo-random numbers.
If two independent random variables
and
have uniform distributions on
,
then their sum has the so-called
triangular distribution
on
with density
for
and
for
.
The sum of three independent random variables with uniform distributions on
has on
the distribution with density
In general, the distribution of the sum
of independent variables with uniform distributions on
has density
for
and
for
;
here
As
,
the distribution of the sum
,
centred around the mathematical expectation
and scaled by the standard deviation
,
tends to the normal distribution with parameters 0 and 1 (the approximation for
is already satisfactory for many practical purposes).
In statistical applications the procedure for constructing a random variable
with given distribution function
is based on the following fact. Let the random variable
be uniformly distributed on
and let the distribution function
be continuous and strictly increasing. Then the random variable
has distribution function
(in the general case it is necessary to replace the inverse function
in the definition of
by an analogue, namely
).
The uniform distribution on an interval as a limit distribution.
Some typical examples of the uniform distribution on
arising as a limit are given below.
1)
Let
be independent random variables having the same
continuous distribution function. Then the distribution of their sums
,
taken
,
that is, the distribution of the fractional parts
of these sums
,
converges to the uniform distribution on
.
2)
Let the random parameters
and
have an absolutely-continuous joint distribution; then, as
,
the distribution of
converges to the uniform distribution on
.
3)
A uniform distribution appears as the limit
distribution of the fractional parts of certain functions
on the positive integers. For example, for irrational
the fraction of those
,
,
for which
has the limit
as
.
The uniform distribution on subsets of
.
An example of a uniform distribution in a rectangle appears already in the
Buffon problem
(see also
Geometric probabilities;
Stochastic geometry).
The uniform distribution on a bounded set
in
is defined as the distribution with density
where
is the inverse of the
-dimensional
volume (or Lebesgue measure) of
.
Uniform distributions on surfaces have also been discussed.
Thus, a
"random direction"
(for example, in
),
defined as a vector from the origin to a random
point on the surface of the unit sphere, is uniformly distributed
in the sense that the probability that it hits a part
of the surface is proportional to the area of that part.
The role of the uniform distribution in algebraic groups is played by the normalized
Haar measure.