Difference between revisions of "Uniform distribution"
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''2''' , Wiley (1971) | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''2''', Wiley (1971) </TD></TR></table> |
Revision as of 09:22, 4 May 2012
2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]
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.
References
| [1] | W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971) |
Uniform distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_distribution&oldid=25943