Continuous uniform distribution

Properties

Moments

The mean (first raw moment) of the continuous uniform distribution is: $${\displaystyle \mathrm{E}[X]={\int }_a^{b}x\frac{dx}{b-a}=\frac{b^2-a^2}{2(b-a)}=\frac{b+a}{2}.}$$

The second raw moment of this distribution is: $${\displaystyle \mathrm{E}\left[X^2\right]={\int }_a^b{x}^2\frac{dx}{b-a}=\frac{b^3-a^3}{3(b-a)}.}$$

In general, the ${\displaystyle n}$-th raw moment of this distribution is: $${\displaystyle \mathrm{E}\left[X^n\right]={\int }_a^b{x}^n\frac{dx}{b-a}=\frac{b^{n+1}-a^{n+1}}{(n+1)(b-a)}.}$$

The variance (second central moment) of this distribution is: $${\displaystyle \mathrm{Var}[X]=\mathrm{E}\left[{\left(X-\mathrm{E}[X]\right)}^2\right]={\int }_a^b{\left(x-\frac{a+b}{2}\right)}^2\frac{dx}{b-a}=\frac{(b-a{)}^2}{12}.}$$

Order statistics

Let ${\displaystyle X_1,...,X_n}$ be an i.i.d. sample from ${\displaystyle U(0,1),}$ and let ${\displaystyle X_{(k)}}$ be the ${\displaystyle k}$-th order statistic from this sample.

${\displaystyle X_{(k)}}$ has a beta distribution, with parameters ${\displaystyle k}$ and ⁠${\displaystyle n-k+1.}$⁠

The expected value is: $${\displaystyle \mathrm{E}\left[X_{(k)}\right]=\frac{k}{n+1}.}$$

This fact is useful when making Q–Q plots.

The variance is: $${\displaystyle \mathrm{Var}\left[X_{(k)}\right]=\frac{k(n-k+1)}{(n+1{)}^2(n+2)}.}$$

Uniformity

The probability that a continuously uniformly distributed random variable falls within any interval of fixed length is independent of the location of the interval itself (but it is dependent on the interval size ${\displaystyle (\ell )}$), so long as the interval is contained in the distribution's support.

Indeed, if ${\displaystyle X\sim U(a,b)}$ and if ${\displaystyle [x,x+\ell ]}$ is a subinterval of ${\displaystyle [a,b]}$ with fixed ${\displaystyle \ell >0,}$ then: $${\displaystyle Pr(X\in [x,x+\ell ])={\int }_x^{x+\ell }\frac{dy}{b-a}=\frac{\ell }{b-a},}$$ which is independent of ${\displaystyle x.}$ This fact motivates the distribution's name.

Uniform distribution on more general sets

The uniform distribution can be generalized to sets more general than intervals.

Formally, let ${\displaystyle S}$ be a Borel set of positive, finite Lebesgue measure ${\displaystyle \lambda (S),}$ i.e. ${\displaystyle 0<\lambda (S)<+\mathrm{\infty }.}$ The uniform distribution on ${\displaystyle S}$ can be specified by defining the probability density function to be zero outside ${\displaystyle S}$ and constantly equal to ${\displaystyle \frac{1}{\lambda (S)}}$ on ${\displaystyle S.}$

Many special cases are known, e.g., when the set S is a simplex.

Related distributions

• If X has a standard uniform distribution, then by the inverse transform sampling method, Y = − λ⁻¹ ln(X) has an exponential distribution with (rate) parameter λ.
• If X has a standard uniform distribution, then Y = Xⁿ has a beta distribution with parameters (1/n,1). As such,
• The standard uniform distribution is a special case of the beta distribution, with parameters (1,1).
• The Irwin–Hall distribution is the sum of n i.i.d. U(0,1) distributions.
• The Bates distribution is the average of n i.i.d. U(0,1) distributions.
• The sum of two independent uniform distributions U₁(a,b)+U₂(c,d) yields a trapezoidal distribution, symmetric about its mean, on the support [a+c,b+d]. The plateau has width equals to the absolute different of the width of U₁ and U₂. The width of the sloped parts corresponds to the width of the narrowest uniform distribution.
  • If the uniform distributions have the same width w, the result is a triangular distribution, symmetric about its mean, on the support [a+c,a+c+2w].
  • The sum of two independent, equally distributed, uniform distributions U₁(a,b)+U₂(a,b) yields a symmetric triangular distribution on the support [2a,2b].
• The distance between two i.i.d. uniform random variables |U₁(a,b)−U₂(a,b)| also has a triangular distribution, although not symmetric, on the support [0,b−a].

Statistical inference

Estimation of parameters

Estimation of maximum

Minimum-variance unbiased estimator

Given a uniform distribution on ${\displaystyle [0,b]}$ with unknown ${\displaystyle b,}$ the minimum-variance unbiased estimator (UMVUE) for the maximum is: $${\displaystyle {\hat{b}}_{\text{UMVU}}=\frac{k+1}{k}m=m+\frac{m}{k},}$$ where ${\displaystyle m}$ is the sample maximum and ${\displaystyle k}$ is the sample size, sampling without replacement (though this distinction almost surely makes no difference for a continuous distribution). This follows for the same reasons as estimation for the discrete distribution, and can be seen as a very simple case of maximum spacing estimation. This problem is commonly known as the German tank problem, due to application of maximum estimation to estimates of German tank production during World War II.

Method of moments estimator

The method of moments estimator is: $${\displaystyle {\hat{b}}_{MM}=2\overline{X},}$$ where ${\displaystyle \overline{X}}$ is the sample mean.

Maximum likelihood estimator

The maximum likelihood estimator is: $${\displaystyle {\hat{b}}_{ML}=m,}$$ where ${\displaystyle m}$ is the sample maximum, also denoted as ${\displaystyle m=X_{(n)},}$ the maximum order statistic of the sample.

Estimation of minimum

Given a uniform distribution on ${\displaystyle [a,b]}$ with unknown a, the maximum likelihood estimator for a is: $${\displaystyle {\hat{a}}_{ML}=min\{X_1,\dots ,X_n\},}$$ the sample minimum.

Estimation of midpoint

The midpoint of the distribution, ${\displaystyle \frac{a+b}{2},}$ is both the mean and the median of the uniform distribution. Although both the sample mean and the sample median are unbiased estimators of the midpoint, neither is as efficient as the sample mid-range, i.e. the arithmetic mean of the sample maximum and the sample minimum, which is the UMVU estimator of the midpoint (and also the maximum likelihood estimate).

Confidence interval

For the maximum

Let ${\displaystyle X_1,X_2,X_3,...,X_n}$ be a sample from ${\displaystyle U_{[0,L]},}$ where ${\displaystyle L}$ is the maximum value in the population. Then ${\displaystyle X_{(n)}=max(X_1,X_2,X_3,...,X_n)}$ has the Lebesgue–Borel density ${\displaystyle f=\frac{d\underset{X_{(n)}}{Pr}}{d\lambda }:}$

$${\displaystyle f(t)=n\frac{1}{L}{\left(\frac{t}{L}\right)}^{n-1}=n\frac{t^{n-1}}{L^n}11_{[0,L]}(t),}$$ where ${\displaystyle 11_{[0,L]}}$ is the indicator function of ${\displaystyle [0,L].}$

The confidence interval given before is mathematically incorrect, as $${\displaystyle Pr([\hat{\theta },\hat{\theta }+\epsilon ]\ni \theta )\ge 1-\alpha }$$ cannot be solved for ${\displaystyle \epsilon }$ without knowledge of ${\displaystyle \theta }$. However, one can solve $${\displaystyle Pr([\hat{\theta },\hat{\theta }(1+\epsilon )]\ni \theta )\ge 1-\alpha }$$ for ${\displaystyle \epsilon \ge {\alpha }^{-1/n}-1}$ for any unknown but valid ${\displaystyle \theta ;}$ one then chooses the smallest ${\displaystyle \epsilon }$ possible satisfying the condition above. Note that the interval length depends upon the random variable ${\displaystyle \hat{\theta }.}$

Occurrence and applications

The probabilities for uniform distribution function are simple to calculate due to the simplicity of the function form. Therefore, there are various applications that this distribution can be used for as shown below: hypothesis testing situations, random sampling cases, finance, etc. Furthermore, generally, experiments of physical origin follow a uniform distribution (e.g. emission of radioactive particles). However, in any application, there is the unchanging assumption that the probability of falling in an interval of fixed length is constant.

Economics example for uniform distribution

In the field of economics, usually demand and replenishment may not follow the expected normal distribution. As a result, other distribution models are used to better predict probabilities and trends such as Bernoulli process. But according to Wanke (2008), in the particular case of investigating lead-time for inventory management at the beginning of the life cycle when a completely new product is being analyzed, the uniform distribution proves to be more useful. In this situation, other distribution may not be viable since there is no existing data on the new product or that the demand history is unavailable so there isn't really an appropriate or known distribution. The uniform distribution would be ideal in this situation since the random variable of lead-time (related to demand) is unknown for the new product but the results are likely to range between a plausible range of two values. The lead-time would thus represent the random variable. From the uniform distribution model, other factors related to lead-time were able to be calculated such as cycle service level and shortage per cycle. It was also noted that the uniform distribution was also used due to the simplicity of the calculations.

Sampling from an arbitrary distribution

The uniform distribution is 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.

Quantization error

In analog-to-digital conversion, a quantization error occurs. This error is either due to rounding or truncation. When the original signal is much larger than one least significant bit (LSB), the quantization error is not significantly correlated with the signal, and has an approximately uniform distribution. The RMS error therefore follows from the variance of this distribution.

Random variate generation

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

On the other hand, the uniformly distributed numbers are often used as the basis for non-uniform random variate generation.

If ${\displaystyle u}$ is a value sampled from the standard uniform distribution, then the value ${\displaystyle a+(b-a)u}$ follows the uniform distribution parameterized by ${\displaystyle a}$ and ${\displaystyle b,}$ as described above.

History

While the historical origins in the conception of uniform distribution are inconclusive, it is speculated that the term "uniform" arose from the concept of equiprobability in dice games (note that the dice games would have discrete and not continuous uniform sample space). Equiprobability was mentioned in Gerolamo Cardano's Liber de Ludo Aleae, a manual written in 16th century and detailed on advanced probability calculus in relation to dice.