View source for Uniform distribution

{{MSC|60E99}}

[[Category:Distribution theory]]

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952402.png" />, is the [[Probability distribution|probability distribution]] with density

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The concept of a uniform distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952404.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952406.png" />. The distribution function is

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and the characteristic function is

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A random variable with uniform distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u0952409.png" /> can be constructed from a sequence of independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524010.png" /> taking the values 0 and 1 with probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524011.png" />, by putting

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(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524013.png" /> are the digits in the binary expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524014.png" />). The random number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524015.png" /> has a uniform distribution in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524016.png" />. This fact has important statistical applications, see, for example, [[Random and pseudo-random numbers|Random and pseudo-random numbers]].

If two independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524018.png" /> have uniform distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524019.png" />, then their sum has the so-called triangular distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524020.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524024.png" />. The sum of three independent random variables with uniform distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524025.png" /> has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524026.png" /> the distribution with density

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In general, the distribution of the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524028.png" /> of independent variables with uniform distributions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524029.png" /> has density

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for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524033.png" />; here

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As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524035.png" />, the distribution of the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524036.png" />, centred around the mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524037.png" /> and scaled by the standard deviation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524038.png" />, tends to the normal distribution with parameters 0 and 1 (the approximation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524039.png" /> is already satisfactory for many practical purposes).

In statistical applications the procedure for constructing a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524040.png" /> with given distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524041.png" /> is based on the following fact. Let the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524042.png" /> be uniformly distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524043.png" /> and let the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524044.png" /> be continuous and strictly increasing. Then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524045.png" /> has distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524046.png" /> (in the general case it is necessary to replace the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524047.png" /> in the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524048.png" /> by an analogue, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524049.png" />).

==The uniform distribution on an interval as a limit distribution.==
Some typical examples of the uniform distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524050.png" /> arising as a limit are given below.

1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524051.png" /> be independent random variables having the same continuous distribution function. Then the distribution of their sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524052.png" />, taken <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524053.png" />, that is, the distribution of the fractional parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524054.png" /> of these sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524055.png" />, converges to the uniform distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524056.png" />.

2) Let the random parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524058.png" /> have an absolutely-continuous joint distribution; then, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524059.png" />, the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524060.png" /> converges to the uniform distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524061.png" />.

3) A uniform distribution appears as the limit distribution of the fractional parts of certain functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524062.png" /> on the positive integers. For example, for irrational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524063.png" /> the fraction of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524065.png" />, for which

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has the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524067.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524068.png" />.

==The uniform distribution on subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524069.png" />.==
An example of a uniform distribution in a rectangle appears already in the [[Buffon problem|Buffon problem]] (see also [[Geometric probabilities|Geometric probabilities]]; [[Stochastic geometry|Stochastic geometry]]). The uniform distribution on a bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524070.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095240/u09524071.png" /> is defined as the distribution with density

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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 "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]].

====References====
<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>