Uniform distribution (continuous)

From Wikipedia, the free encyclopedia

Uniform
Probability density function Using maximum convention
Cumulative distribution function
Parameters	$a,b \in (-\infty,\infty)$
Support	$a \le x \le b$
Probability density function (pdf)	$\begin{matrix} \frac{1}{b - a} & \mbox{for }a \le x \le b \\ \\ 0 & \mathrm{for}\ x<a\ \mathrm{or}\ x>b \end{matrix}$
Cumulative distribution function (cdf)	$\begin{matrix} 0 & \mbox{for }x < a \\ \frac{x-a}{b-a} & ~~~~~ \mbox{for }a \le x < b \\ 1 & \mbox{for }x \ge b \end{matrix}$
Mean	$\frac{a+b}{2}$
Median	$\frac{a+b}{2}$
Mode	any value in $[a, b]$
Variance	$\frac{(b-a)^2}{12}$
Skewness	$0$
Excess Kurtosis	$-\frac{6}{5}$
Entropy	$ln(b - a)$
mgf	$\frac{e^{tb}-e^{ta}}{t(b-a)}$
Char. func.	$\frac{e^{itb}-e^{ita}}{it(b-a)}$

In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable.

The continuous uniform distribution is closely related to the rectangle function because of the shape of its probability density function. It is parametrised by its minimum and maximum values, a and b. The probability density function of the uniform distribution is thus:

$f(x)=\left\{\begin{matrix} \frac{1}{b - a} & \ \ \ \mbox{for }a < x < b, \\ \\ 0 & \mathrm{for}\ x<a\ \mathrm{or}\ x>b, \\ \\ \mathrm{see}\ \mathrm{below} & \ \ \ \mbox{for }x=a \mbox{ or }x=b. \end{matrix}\right.$

The values at the two boundaries a and b are usually unimportant because they do not alter the values of the integrals of f(x) dx over any interval, nor of x f(x) dx or the like. Sometimes they are chosen to be zero, and sometimes chosen to be 1/(b − a). The latter is appropriate in the context of estimation by the method of maximum likelihood. In the context of Fourier analysis, one may take the value of f(a) or f(b) to be 1/(2(b − a)), since then the inverse transform of many integral transforms of this uniform function will yield back the function itself, rather than a function which is equal "almost everywhere", i.e. except on a set of points with zero measure. Also, it is consistent with the sign function which has no such ambiguity.

The distribution is often abbreviated U(a,b).

1 Properties
2 'Uniformity'
3 Standard uniform
4 Derived distributions
5 Relationship to other functions
6 Applications
- 6.1 Sampling from a uniform distribution
- 6.2 Sampling from an arbitrary distribution

[edit] Properties

[edit] Cumulative distribution function

The cumulative distribution function is:

$F(x)=\left\{\begin{matrix} 0 & \mbox{for }x < a \\ \frac{x-a}{b-a} & \ \ \ \mbox{for }a \le x < b \\ 1 & \mbox{for }x \ge b \end{matrix}\right.$

[edit] Moment-generating function

The moment-generating function is

$M_x = E(e^{tx}) = \frac{e^{tb}-e^{ta}}{t(b-a)} \,\!$

from which we may calculate the raw moments m _k

$m_1=\frac{a+b}{2}, \,\!$

$m_2=\frac{a^2+ab+b^2}{3}, \,\!$

$m_k=\frac{1}{k+1}\sum_{i=0}^k a^ib^{k-i}. \,\!$

For a random variable following this distribution, the expected value is then m₁ = (a + b)/2 and the variance is m₂ − m₁² = (b − a)²/12.

This distribution can be generalized to more complicated sets than intervals. If S is a Borel set of positive, finite measure, the uniform probability distribution on S can be specified by saying that the pdf is zero outside S and constantly equal to 1/K on S, where K is the Lebesgue measure of S.

For n ≥ 2, the nth cumulant of the uniform distribution on the interval [0, 1] is b_b/n, where b_n is the nth Bernoulli number.

[edit] Order statistics

Let X₁, ..., X_n be an i.i.d. sample from U(0,1). Let X_(k) be the kth order statistic from this sample. Then the probability distribution of X_(k) is a Beta distribution with parameters k and n − k + 1. The expected value is

$\operatorname{E}(X_{(k)}) = {k \over n+1}.$

This fact is useful when making Q-Q plots.

Moreover, if X₁, ..., X_n is an i.i.d. sample from U(0,θ). The expected values are

$\operatorname{E}(X_{(k)}) = {k \theta \over n+1} .$

The variances are

$\operatorname{Var}(X_{(k)}) = {k (n-k+1) \theta^2 \over (n+1)^2 (n+2)} .$

[edit] 'Uniformity'

The probability that a uniformly distributed random variable falls within any interval of fixed length is independent of the interval itself, so long as it is fully within the distribution's support.

To see this, if X ≈ U(a,b) and [x, x+d] is a subinterval of [a,b] with fixed d > 0, then

$P\left(X\in\left [ x,x+d \right ]\right) = \int_{x}^{x+d} \frac{\mathrm{d}y}{b-a}\, = \frac{d}{b-a} \,\!$

which is independent of x. This fact motivates the distribution's name.

[edit] Standard uniform

Restricting $a = 0$ and $b = 1$ , the resulting distribution U(0,1) is called a standard uniform distribution.

One interesting property of the standard uniform distribution is that if u₁ has a standard uniform distribution, then so does 1-u₁.

[edit] Derived distributions

If X has a standard uniform distribution,

Y = -ln(X)/λ has an exponential distribution with (rate) parameter λ.
Y = 1 - X^1/n has a beta distribution with parameters 1 and n. (Note this implies that the standard uniform distribution is a special case of the beta distribution, with parameters 1 and 1.)

[edit] Relationship to other functions

As long as the same conventions are followed at the transition points, the probability density function may also be expressed in terms of the Heaviside step function:

$f(x)=\frac{\operatorname{H}(x-a)-\operatorname{H}(x-b)}{b-a}, \,\!$

or in terms of the rectangle function

$f(x)=\frac{1}{b-a}\,\operatorname{rect}\left(\frac{x-\left(\frac{a+b}{2}\right)}{b-a}\right) \,\!$

There is no ambiguity at the transition point of the sign function. Using the half-maximum convention at the transition points, the uniform distribution may be expressed in terms of the sign function as:

$f(x)=\frac{ \sgn{(x-a)}-\sgn{(x-b)}} {2(b-a)} \,\!$

[edit] Applications

In statistics, when a p-value is used as a test statistic for a simple null hypothesis, and the distribution of the test statistic is continuous, then the test statistic is uniformly distributed between 0 and 1 if the null hypothesis is true.

[edit] Sampling from a uniform distribution

There are many applications in which it is useful to run simulation experiments. Many programming languages have the ability to generate pseudo-random numbers which are effectively distributed according to the standard uniform distribution.

If u is a value sampled from the standard uniform distribution, then the value a + (b − a)u follows the uniform distribution parametrised by a and b, as described above.

[edit] Sampling from an arbitrary distribution

Although the uniform distribution is not commonly found in nature, it is particularly useful for sampling from arbitrary distributions. A general method is the inverse transform sampling method, which uses the cumulative distribution function (CDF) of the target random variable. This method is very useful in theoretical work. Since simulations using this method require inverting the CDF of the target variable, alternative methods have been devised for the cases where the CDF is not known in closed form. One such method is rejection sampling.

The normal distribution is an important example where the inverse transform method is not efficient. However, there is an exact method, the Box-Muller transformation, which uses the inverse transform to convert two independent uniform random variables into two independent normally distributed random variables.

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse Gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular