Uniform distribution

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 $ [a,\ b] $, $ a < b $, is the probability distribution with density $$ p (x) = \left \{ \begin{array}{ll} { \frac{1}{b - a} } , & x \in [a,\ b], \\ 0, & x \notin [a,\ b]. \\ \end{array} \right .$$ The concept of a uniform distribution on $ [a,\ b] $ 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 $ (b + a)/2 $ and $ (b - a) ^{2} /12 $. The distribution function is $$ F (x) = \left \{ \begin{array}{ll} 0 , & x \leq a, \\ \frac{x - a}{b - a} , & a < x \leq b, \\ 1, & x > b, \\ \end{array} \right .$$ and the characteristic function is $$ \phi (t) = { \frac{1}{it (b - a)} } (e ^{itb} - e ^{ita} ). $$ A random variable with uniform distribution on $ [0,\ 1] $ can be constructed from a sequence of independent random variables $ X _{1} ,\ X _{2} \dots $ taking the values 0 and 1 with probabilities $ 1/2 $, by putting $$ X = \sum _ {n = 1} ^ \infty X _{n} 2 ^{-n} $$( $ X _{n} $ are the digits in the binary expansion of $ X $). The random number $ X $ has a uniform distribution in the interval $ [0,\ 1] $. This fact has important statistical applications, see, for example, Random and pseudo-random numbers.

If two independent random variables $ X _{1} $ and $ X _{2} $ have uniform distributions on $ [0,\ 1] $, then their sum has the so-called triangular distribution on $ [0,\ 2] $ with density $ u _{2} (x) = 1 - | 1 - x | $ for $ x \in [0,\ 2] $ and $ u _{2} (x) = 0 $ for $ x \notin [0,\ 2] $. The sum of three independent random variables with uniform distributions on $ [0,\ 1] $ has on $ [0,\ 3] $ the distribution with density $$ u _{3} (x) = \left \{ \begin{array}{ll} { \frac{x ^ 2}{2} } , & 0 \leq x < 1, \\ { \frac{[x ^{2} - 3 (x - 1) ^{2} ]}{2} } , & 1 \leq x < 2, \\ { \frac{[x ^{2} - 3 (x - 1) ^{2} + 3 (x - 2) ^{2} ]}{2} } , & 2 \leq x < 3, \\ 0, & x \notin [0,\ 3]. \\ \end{array} \right .$$ In general, the distribution of the sum $ X _{1} + \dots + X _{n} $ of independent variables with uniform distributions on $ [0,\ 1] $ has density $$ u _{n} (x) = { \frac{1}{(n - 1)!} } \sum _ {k = 0} ^ n (-1) ^{k} \binom{n}{k} (x - k) _{+} ^ {n - 1} $$ for $ 0 \leq x \leq n $ and $ u _{n} (x) = 0 $ for $ x \notin [0,\ n] $; here $$ z _{+} = \left \{ \begin{array}{ll} z, & z > 0, \\ 0, & z \leq 0. \\ \end{array} \right .$$ As $ n \rightarrow \infty $, the distribution of the sum $ X _{1} + \dots + X _{n} $, centred around the mathematical expectation $ n/2 $ and scaled by the standard deviation $ \sqrt {n/12} $, tends to the normal distribution with parameters 0 and 1 (the approximation for $ n = 3 $ is already satisfactory for many practical purposes).

In statistical applications the procedure for constructing a random variable $ X $ with given distribution function $ F $ is based on the following fact. Let the random variable $ Y $ be uniformly distributed on $ [0,\ 1] $ and let the distribution function $ F $ be continuous and strictly increasing. Then the random variable $ X = F ^ {\ -1} Y $ has distribution function $ F $( in the general case it is necessary to replace the inverse function $ F ^ {\ -1} (y) $ in the definition of $ X $ by an analogue, namely $ F ^ {\ -1} (y) = \mathop{\rm inf}\nolimits \{ {x} : {F (x) \leq y \leq F (x + 0)} \} $).

The uniform distribution on an interval as a limit distribution.

Some typical examples of the uniform distribution on $ [0,\ 1] $ arising as a limit are given below.

1) Let $ X _{1} ,\ X _{2} \dots $ be independent random variables having the same continuous distribution function. Then the distribution of their sums $ S _{n} $, taken $ \mathop{\rm mod}\nolimits \ 1 $, that is, the distribution of the fractional parts $ \{ S _{n} \} $ of these sums $ S _{n} $, converges to the uniform distribution on $ [0,\ 1] $.

2) Let the random parameters $ \alpha $ and $ \beta $ have an absolutely-continuous joint distribution; then, as $ t \rightarrow \infty $, the distribution of $ \{ \alpha t + \beta \} $ converges to the uniform distribution on $ [0,\ 1] $.

3) A uniform distribution appears as the limit distribution of the fractional parts of certain functions $ g $ on the positive integers. For example, for irrational $ \alpha $ the fraction of those $ m $, $ 1 \leq m \leq n $, for which $$ 0 \leq a \leq \{ na \} \leq b \leq 1, $$ has the limit $ b - a $ as $ n \rightarrow \infty $.

The uniform distribution on subsets of $ \mathbf R ^{k} $.

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 $ D $ in $ \mathbf R ^{k} $ is defined as the distribution with density $$ p (x _{1} \dots x _{n} ) = \left \{ \begin{array}{ll} C \neq 0, & x \in D, \\ 0, & x \notin D, \\ \end{array} \right .$$ where $ C $ is the inverse of the $ k $- dimensional volume (or Lebesgue measure) of $ D $.

Uniform distributions on surfaces have also been discussed. Thus, a "random direction" (for example, in $ \mathbf R ^{3} $), 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

[F]	W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)