Difference between revisions of "Uniform distribution"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
||
Line 1: | Line 1: | ||
− | A common name for a class of probability distributions, arising as an extension of the idea of | + | 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|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 of the line (the rectangular distribution).== | ||
Line 56: | Line 56: | ||
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524073.png" /> is the inverse of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524074.png" />-dimensional volume (or Lebesgue measure) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524075.png" />. | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524073.png" /> is the inverse of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524074.png" />-dimensional volume (or Lebesgue measure) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524075.png" />. | ||
− | Uniform distributions on surfaces have also been discussed. Thus, a | + | Uniform distributions on surfaces have also been discussed. Thus, a "random direction" (for example, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524076.png" />), 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|Haar measure]]. | The role of the uniform distribution in algebraic groups is played by the normalized [[Haar measure|Haar measure]]. | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''2''' , Wiley (1971) {{MR|0270403}} {{ZBL|0219.60003}} </TD></TR></table> |
Revision as of 14:52, 24 March 2012
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) MR0270403 Zbl 0219.60003 |
Uniform distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_distribution&oldid=21956