View source for Continuous uniform distribution

{{Short description|Uniform distribution on an interval}}
{{Probability distribution
  |name       = Continuous uniform
  |type       = density
  |pdf_image  = [[Image:Uniform Distribution PDF SVG.svg|250px|PDF of the uniform probability distribution using [[Heaviside step function#Zero argument|the maximum convention]] at the transition points.]]<BR /><small>Using [[Heaviside step function#Zero argument|maximum convention]]</small>|
  |cdf_image  = [[Image:Uniform cdf.svg|250px|CDF of the uniform probability distribution.]]|
  |notation   = <math>\mathcal{U} _{[a,b]}</math>  
  |parameters = <math>- \infty < a < b < \infty</math>
  |support    = <math>[a,b]</math>
  |pdf        = <math>\begin{cases}
                  \frac{1}{b-a}   & \text{for } x \in [a,b] \\
                  0               & \text{otherwise}
                \end{cases}</math>
  |cdf        = <math>\begin{cases}
                  0               & \text{for } x < a \\
                  \frac{x-a}{b-a} & \text{for } x \in [a,b] \\
                  1               & \text{for } x > b
                \end{cases}</math>
  |mean       = <math>\tfrac{1}{2}(a+b)</math>
  |median     = <math>\tfrac{1}{2}(a+b)</math>
  |mode       = <math>\text{any value in } (a,b)</math>
  |variance   = <math>\tfrac{1}{12} (b-a)^2</math>
  |mad        = <math>\tfrac{1}{4} (b-a)</math>
  |skewness   = <math>0</math>
  |kurtosis   = <math>- \tfrac{6}{5}</math>
  |entropy    = <math>\log (b-a)</math>
  |mgf        = <math>\begin{cases}
                        \frac{ \mathrm{e} ^{tb} - \mathrm{e} ^{ta} }{t(b-a)} & \text{for } t \neq 0 \\
                         1 & \text{for } t = 0
                      \end{cases}</math>
  |char       = <math>\begin{cases}
                        \frac{ \mathrm{e} ^{ \mathrm{i} tb } - \mathrm{e} ^{ \mathrm{i} ta } }{ \mathrm{i} t(b-a) } & \text{for } t \neq 0 \\
                         1 & \text{for } t = 0
                      \end{cases}</math>
}}

In [[probability theory]] and [[statistics]], the '''continuous uniform distributions''' or '''rectangular distributions''' are a family of [[Symmetric distribution|symmetric]] [[probability distributions]]. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.<ref name=":0">{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|url=https://archive.org/details/modernintroducti00fmde|url-access=limited|last=Dekking|first=Michel|publisher=Springer|year=2005|isbn=978-1-85233-896-1|location=London, UK|pages=[https://archive.org/details/modernintroducti00fmde/page/n71 60]–61}}</ref> The bounds are defined by the parameters, <math>a</math> and <math>b,</math> which are the minimum and maximum values. The interval can either be [[Closed interval|closed]] (i.e. <math>[a,b]</math>) or [[Open interval|open]] (i.e. <math>(a,b)</math>).<ref name=":3">{{Cite book|title=Probability & Statistics for Engineers and Scientists|last=Walpole |display-authors=etal |first=Ronald|publisher=Prentice Hall|year=2012|isbn=978-0-321-62911-1|location=Boston, USA|pages=171–172}}</ref> Therefore, the distribution is often abbreviated <math>U(a,b),</math> where <math>U</math> stands for uniform distribution.<ref name=":0" /> The difference between the bounds defines the interval length; all [[Interval (mathematics)|interval]]s of the same length on the distribution's [[Support (mathematics)|support]] are equally probable. It is the [[maximum entropy probability distribution]] for a [[random variable]] <math>X</math> under no constraint other than that it is contained in the distribution's support.<ref>{{cite journal |last1=Park |first1=Sung Y. |last2=Bera |first2=Anil K. |year=2009 |title=Maximum entropy autoregressive conditional heteroskedasticity model |journal=[[Journal of Econometrics]] |volume=150 |issue=2 |pages=219–230 |doi=10.1016/j.jeconom.2008.12.014 |citeseerx=10.1.1.511.9750 }}</ref>

== Definitions ==
=== Probability density function ===
The [[probability density function]] of the continuous uniform distribution is
<math display="block">
  f(x) = \begin{cases}
  \dfrac{1}{b-a} & \text{for } a \le x \le b, \\[8pt]
  0 & \text{for } x < a \ \text{ or } \ x > b.
  \end{cases}
</math>

The values of <math>f(x)</math> at the two boundaries <math>a</math> and <math>b</math> are usually unimportant, because they do not alter the value of <math display="inline">\int_c^d f(x)dx</math> over any interval <math>[c,d],</math> nor of <math display="inline">\int_a^b x f(x) \, dx,</math> nor of any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be <math>\tfrac{1}{b-a} .</math> 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 <math>f(a)</math> or <math>f(b)</math> to be <math>\tfrac{1}{2(b-a)} ,</math> because then the inverse transform of many [[integral transform]]s 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 theory|measure]]. Also, it is consistent with the [[sign function]], which has no such ambiguity.

Any probability density function integrates to <math>1,</math> so the probability density function of the continuous uniform distribution is graphically portrayed as a rectangle where {{tmath|b-a}} is the base length and {{tmath|\tfrac{1}{b-a} }} is the height. As the base length increases, the height (the density at any particular value within the distribution boundaries) decreases.<ref name=":1">{{Cite web|url=https://www.mathworks.com/help/stats/uniform-distribution-continuous.html|title=Uniform Distribution (Continuous)|date=2019|website=MathWorks|access-date=November 22, 2019}}</ref>

In terms of mean <math>\mu</math> and variance <math>\sigma ^2 ,</math> the probability density function of the continuous uniform distribution is
<math display="block">
 f(x) = \begin{cases}
 \dfrac{1}{2 \sigma \sqrt{3}} & \text{for } - \sigma \sqrt{3} \le x - \mu \le \sigma \sqrt{3} , \\[2pt]
 0 & \text{otherwise} .
 \end{cases}
</math>

=== Cumulative distribution function ===
The [[cumulative distribution function]] of the continuous uniform distribution is:
<math display="block"> F(x) = \begin{cases}
  0 & \text{for } x < a, \\[8pt]
  \frac{x-a}{b-a} & \text{for } a \le x \le b, \\[8pt]
  1 & \text{for } x > b.
\end{cases} </math>

Its inverse is:
<math display="block">F^{-1} (p) = a + p (b - a) \quad \text{ for } 0 < p < 1.</math>

In terms of mean <math>\mu</math> and variance <math>\sigma ^2 ,</math> the cumulative distribution function of the continuous uniform distribution is:
<math display="block">F(x) = \begin{cases}
0 & \text{for } x - \mu < - \sigma \sqrt{3} , \\
\frac{1}{2} \left( \frac{x - \mu}{ \sigma \sqrt{3} } + 1 \right) & \text{for } - \sigma \sqrt{3} \le x - \mu < \sigma \sqrt{3} , \\
1 & \text{for } x - \mu \ge \sigma \sqrt{3} ;
\end{cases}</math>

its inverse is:
<math display="block">F^{-1} (p) = \sigma \sqrt{3} (2p-1) + \mu \quad \text{ for } 0 \le p \le 1.</math>

==== <u>Example 1.</u> Using the continuous uniform distribution function ====
For a [[random variable]] <math>X \sim U(0,23),</math> find <math>\Pr(2 < X < 18):</math>
<math display="block">\Pr(2 < X < 18) = (18-2) \cdot \frac{1}{23-0} = \frac{16}{23} .</math>

In a graphical representation of the continuous uniform distribution function <math>[f(x) \text{ vs } x],</math> the area under the curve within the specified bounds, displaying the probability, is a rectangle. For the specific example above, the base would be {{tmath|16,}} and the height would be {{tmath|\tfrac{1}{23} .}}<ref name=":2">{{Cite book|title=Introductory Statistics|url=https://archive.org/details/IntroductoryStatistics_201904|last=Illowsky |display-authors=etal |first=Barbara|publisher=OpenStax College|year=2013|isbn=978-1-938168-20-8|location=Rice University, Houston, Texas, USA|pages=[https://archive.org/details/IntroductoryStatistics_201904/page/n303 296]–304}}</ref>

==== <u>Example 2.</u> Using the continuous uniform distribution function (conditional) ====
For a random variable <math>X \sim U(0,23),</math> find <math>\Pr(X > 12 \mid X > 8):</math>
<math display="block">\Pr(X > 12 \mid X > 8) = (23-12) \cdot \frac{1}{23-8} = \frac{11}{15}.</math>

The example above is a [[conditional probability]] case for the continuous uniform distribution: given that {{tmath|X > 8}} is true, what is the probability that {{tmath|X > 12?}} Conditional probability changes the sample space, so a new interval length {{tmath|b-a'}} has to be calculated, where <math>b = 23</math> and <math>a' = 8.</math><ref name=":2" /> The graphical representation would still follow Example 1, where the area under the curve within the specified bounds displays the probability; the base of the rectangle would be {{tmath|11,}} and the height would be {{tmath|\tfrac{1}{15}.}}<ref name=":2" />

=== Generating functions ===
==== Moment-generating function ====
The [[moment-generating function]] of the continuous uniform distribution is:<ref>{{harvnb|Casella|Berger|2001|page=626}}</ref>
<math display="block">M_X = \operatorname{E}\left[ e^{tX} \right] = \int_a^b e^{tx} \frac{dx}{b-a}
= \frac{ e^{tb} - e^{ta} }{t(b-a)} = \frac{B^t - A^t}{t(b-a)} ,</math>
from which we may calculate the [[raw moments]] <math>m_k :</math>
<math display="block">m_1 = \frac{a+b}{2} ,</math>
<math display="block">m_2 = \frac{a^2+ab+b^2}{3} ,</math>
<math display="block">m_k = \frac{ \sum_{i=0}^k a^i b^{k-i} }{k+1} .</math>

For a random variable following the continuous uniform distribution, the [[expected value]] is <math>m_1 = \tfrac{a+b}{2} ,</math> and the [[variance]] is <math>m_2 - m_1 ^2 = \tfrac{(b-a)^2}{12} .</math>

For the special case <math>a = -b,</math> the probability density function of the continuous uniform distribution is:
<math display="block">
  f(x) = \begin{cases}
  \frac{1}{2b} & \text{for } -b \le x \le b, \\[8pt]
  0 & \text{otherwise} ;
  \end{cases}
</math>

the moment-generating function reduces to the simple form:
<math display="block">M_X = \frac{ \sinh bt }{bt} .</math>

==== Cumulant-generating function ====
For {{tmath|n \ge 2,}} the <math>n</math>-th [[cumulant]] of the continuous uniform distribution on the interval {{tmath|[- \tfrac{1}{2} , \tfrac{1}{2}]}} is <math>\tfrac{B_n}{n} ,</math> where <math>B_n</math> is the <math>n</math>-th [[Bernoulli number]].<ref>{{cite web|url=https://galton.uchicago.edu/~wichura/Stat304/Handouts/L18.cumulants.pdf|title=Cumulants|date=January 11, 2001|work=Stat 304 Handouts|first=Michael J.|last=Wichura|publisher=University of Chicago}}</ref>

=== Standard uniform distribution ===
The continuous uniform distribution with parameters <math>a = 0</math> and <math>b = 1,</math> i.e. <math>U(0,1),</math> is called the '''standard uniform distribution'''.

One interesting property of the standard uniform distribution is that if <math>u_1</math> has a standard uniform distribution, then so does <math>1 - u_1 .</math> This property can be used for generating [[antithetic variates]], among other things. In other words, this property is known as the [[inversion method]] where the continuous standard uniform distribution can be used to generate [[Statistical randomness|random numbers]] for any other continuous distribution.<ref name=":1" /> If <math>u_1</math> is a uniform random number with standard uniform distribution, i.e. with <math>U(0,1),</math> then <math>x = F^{-1} (u_1)</math> generates a random number <math>x</math> from any continuous distribution with the specified [[cumulative distribution function]] <math>F.</math><ref name=":1" />

=== Relationship to other functions ===
As long as the same conventions are followed at the transition points, the probability density function of the continuous uniform distribution may also be expressed in terms of the [[Heaviside step function]] as:
<math display="block">f(x) = \frac{ \operatorname{H} (x-a) - \operatorname{H} (x-b) }{b-a} ,</math>

or in terms of the [[rectangle function]] as:
<math display="block">f(x) = \frac{1}{b-a} \ \operatorname{rect} \left( \frac{x - \frac{a+b}{2}}{b-a} \right) .</math>

There is no ambiguity at the transition point of the [[sign function]]. Using the half-maximum convention at the transition points, the continuous uniform distribution may be expressed in terms of the sign function as:
<math display="block">f(x) = \frac{ \sgn{(x-a)} - \sgn{(x-b)} }{2(b-a)} .</math>

== Properties ==
=== Moments ===
The mean (first raw [[Moment (mathematics)|moment]]) of the continuous uniform distribution is:
<math display="block">\operatorname{E}[X] = \int_a^b x \frac{dx}{b-a} = \frac{b^2 - a^2}{2 (b-a)} = \frac{b + a}{2} .</math>

The second raw moment of this distribution is:
<math display="block">\operatorname{E}\left[X^2\right] = \int_a^b x^2 \frac{dx}{b-a} = \frac{b^3 - a^3}{3(b-a)} .</math>

In general, the <math>n</math>-th raw moment of this distribution is:
<math display="block">\operatorname{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)} .</math>

The variance (second [[central moment]]) of this distribution is:
<math display="block">\operatorname{Var}[X] = \operatorname{E}\left[ {\left( X - \operatorname{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} .</math>

=== Order statistics ===
Let <math>X_1 , ..., X_n</math> be an [[i.i.d.]] sample from <math>U(0,1),</math> and let <math>X_{(k)}</math> be the <math>k</math>-th [[order statistic]] from this sample.

<math>X_{(k)}</math> has a [[beta distribution]], with parameters <math>k</math> and {{tmath|n-k+1.}}

The expected value is:
<math display="block">\operatorname{E} \left[X_{(k)}\right] = \frac{k}{n+1} .</math>

This fact is useful when making [[Q–Q plot]]s.

The variance is:
<math display="block">\operatorname{Var} \left[X_{(k)}\right] = \frac{k(n-k+1)}{(n+1)^2 (n+2)} .</math>

{{See also|Order statistic#Probability distributions of order statistics}}

=== 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 <math>( \ell )</math>), so long as the interval is contained in the distribution's support.

Indeed, if <math>X \sim U(a,b)</math> and if <math>[x,x+ \ell ]</math> is a subinterval of <math>[a,b]</math> with fixed <math>\ell > 0,</math> then:
<math display="block">
  \Pr \big( X \in [x, x + \ell ] \big)
  = \int_x^{x+ \ell} \frac{dy}{b-a}
  = \frac{ \ell }{b-a} ,
</math>
which is independent of <math>x.</math> 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 <math>S</math> be a [[Borel set]] of positive, finite [[Lebesgue measure]] <math>\lambda (S),</math> i.e. <math>0 < \lambda (S) < + \infty .</math> The uniform distribution on <math>S</math> can be specified by defining the probability density function to be zero outside <math>S</math> and constantly equal to <math>\tfrac{1}{\lambda (S)}</math> on <math>S.</math>

An interesting special case is when the set S is a '''[[simplex]]'''. It is possible to obtain a uniform distribution on the standard ''n''-vertex simplex in the following way.<ref>{{Cite book |url=https://link.springer.com/book/10.1007/978-1-4613-8643-8 |title=Non-Uniform Random Variate Generation |language=en |doi=10.1007/978-1-4613-8643-8}}</ref>{{Rp|location=Thm.4.1}}take ''n'' independent random variables with the same [[exponential distribution]]; denote them by X<sub>1</sub>,...,X<sub>n</sub>; and let Y<sub>i</sub> := X<sub>i</sub> / (sum<sub>i</sub> X<sub>i</sub>). Then, the vector Y<sub>1</sub>,...,Y<sub>n</sub> is uniformly distributed on the simplex.

== Related distributions ==
* If ''X'' has a standard uniform distribution, then by the [[inverse transform sampling]] method, ''Y'' = − ''λ''<sup>−1</sup> ln(''X'') has an [[exponential distribution]] with (rate) parameter ''λ''.
* If ''X'' has a standard uniform distribution, then ''Y'' = ''X''<sup>''n''</sup> has a [[beta distribution]] with parameters (1/''n'',1). As such,
* 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 standard uniform distribution is a special case of the [[beta distribution]], with parameters (1,1).
* The sum of two independent uniform distributions ''U''<sub>1</sub>(a,b)+''U''<sub>2</sub>(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 ''U''<sub>1</sub> and ''U''<sub>2</sub>. 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 ''U''<sub>1</sub>(a,b)+''U''<sub>2</sub>(a,b) yields a symmetric [[triangular distribution]] on the support [2a,2b].
* The distance between two [[i.i.d.]] uniform random variables |''U''<sub>1</sub>(a,b)-''U''<sub>2</sub>(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 =====
{{Main|German tank problem}}
Given a uniform distribution on <math>[0,b]</math> with unknown <math>b,</math> the [[UMVU|minimum-variance unbiased estimator (UMVUE)]] for the maximum is:
<math display="block">\hat{b} _\text{UMVU} = \frac{k+1}{k} m = m + \frac{m}{k} ,</math>
where <math>m</math> is the [[sample maximum]] and <math>k</math> 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 [[Uniform distribution (discrete)#Estimation of maximum|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 (statistics)|method of moments]] estimator is:
<math display="block">\hat{b} _{MM} = 2 \bar{X} ,</math>
where <math>\bar{X}</math> is the sample mean.

===== Maximum likelihood estimator =====
The [[maximum likelihood]] estimator is:
<math display="block">\hat{b} _{ML} = m ,</math>
where <math>m</math> is the [[sample maximum]], also denoted as <math>m = X_{(n)} ,</math> the maximum [[order statistic]] of the sample.

==== Estimation of minimum ====
Given a uniform distribution on <math>[a,b]</math> with unknown ''a'', the maximum likelihood estimator for ''a'' is:
<math display="block">\hat a_{ML}=\min\{X_1,\dots,X_n\}</math>,
the [[sample minimum]].<ref><math>L_n(a,b)=\prod_{i=1}^n f(X_i)=\frac{1}{(b-a)^n}\mathbf{1}_{[a,b]}(X_1,\dots,X_n)</math><br>
<math>=\frac{1}{(b-a)^n}\mathbf{1}_{\{a\leq\min\{X_1,\dots,X_n\}\}}\mathbf{1}_{\{\max\{X_1,\dots,X_n\}\leq b\}}</math>
.<br>Since we have <math>n\geq 1</math> the factor <math>\frac{1}{(b-a)^n}</math> is maximized by biggest possible ''a'', which is limited in <math>L_n(a,b)</math> by <math>\min\{X_1,\dots,X_n\}</math>. Therefore <math>a=\min\{X_1,\dots,X_n\}</math> is the maximum of <math>L_n(a,b)</math>.</ref>

==== Estimation of midpoint ====
The midpoint of the distribution, <math>\tfrac{a+b}{2} ,</math> is both the mean and the median of the uniform distribution. Although both the sample mean and the sample median are [[unbiased estimator]]s of the midpoint, neither is as [[Efficiency (statistics)|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 <math>X_1 , X_2 , X_3 , ..., X_n</math> be a sample from <math>U_{[0,L]} ,</math> where <math>L</math> is the maximum value in the population. Then <math>X_{(n)} = \max ( X_1 , X_2 , X_3 , ..., X_n )</math> has the Lebesgue–Borel density <math>f = \frac{ d \Pr _{X_{(n)}} }{ d \lambda } :</math><ref name=Nechval2002>Nechval KN, Nechval NA, Vasermanis EK, Makeev VY (2002) [https://www.researchgate.net/profile/Konstantin_Nechval/publication/267986471_CONSTRUCTING_SHORTEST-LENGTH_CONFIDENCE_INTERVALS/links/55c37b3008aeb97567400f29.pdf Constructing shortest-length confidence intervals]. Transport and Telecommunication 3 (1) 95-103</ref>

<math display="block">f(t) = n \frac{1}{L} \left( \frac{t}{L} \right) ^{n-1} \! = n \frac{ t^{n-1} }{ L^n } 1 \! \! 1 _{[0,L]} (t),</math> where <math>1 \! \! 1 _{[0,L]}</math> is the [[indicator function]] of <math>[0,L] .</math>

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

== Occurrence and applications ==
The probabilities for uniform distribution function are simple to calculate due to the simplicity of the function form.<ref name=":3" /> 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 [[Radioactive decay|particles]]).<ref name=":0" /> However, it is important to note that in any application, there is the unchanging assumption that the probability of falling in an interval of fixed length is constant.<ref name=":3" />

=== Economics example for uniform distribution ===
In the field of economics, usually [[demand]] and [[wikt:Special:Search/replenishment|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]].<ref name=":4">{{Cite journal|last=Wanke|first=Peter|date=2008|title=The uniform distribution as a first practical approach to new product inventory management|url=https://www.researchgate.net/publication/223280031|journal=International Journal of Production Economics|volume=114|issue=2|pages=811–819|via=Research Gate|doi=10.1016/j.ijpe.2008.04.004}}</ref> But according to Wanke (2008), in the particular case of investigating [[Lead time|lead-time]] for inventory management at the beginning of the [[Life-cycle assessment|life cycle]] when a completely new product is being analyzed, the uniform distribution proves to be more useful.<ref name=":4" /> 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.<ref name=":4" /> 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.<ref name=":4" /> The [[Lead time|lead-time]] would thus represent the random variable. From the uniform distribution model, other factors related to [[Lead time|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.<ref name=":4" />

=== Sampling from an arbitrary distribution ===
{{Main|Inverse transform sampling}}
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 variable]]s into two independent [[Normal distribution|normally distributed]] random variables.

=== Quantization error ===
{{Main|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|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 language]]s come with implementations to generate [[Pseudorandom number sequence|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 <math>u</math> is a value sampled from the standard uniform distribution, then the value <math>a+(b-a)u</math> follows the uniform distribution parameterized by <math>a</math> and <math>b,</math> 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 uniform distribution|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.<ref>{{Cite journal|last=Bellhouse|first=David|date=May 2005|title=Decoding Cardano's Liber de Ludo|journal=Historia Mathematica|volume=32|pages=180–202|doi=10.1016/j.hm.2004.04.001|doi-access=free}}</ref>

== See also ==
* [[Discrete uniform distribution]]
* [[Beta distribution]]
* [[Box–Muller transform]]
* [[Probability plot (disambiguation)|Probability plot]]
* [[Q–Q plot]]
* [[Rectangular function]]
* [[Irwin–Hall distribution]] — In the degenerate case where n=1, the Irwin-Hall distribution generates a uniform distribution between 0 and 1.
* [[Bates distribution]] — Similar to the Irwin-Hall distribution, but rescaled for n. Like the Irwin-Hall distribution, in the degenerate case where n=1, the Bates distribution generates a uniform distribution between 0 and 1.

== References ==
{{Reflist}}

== Further reading ==
* {{Citation
  | last = Casella
  | first = George
  | author2 = Roger L. Berger
  | title = Statistical Inference
  | year = 2001
  | publisher = Thomson Learning
 | edition = 2nd
  | isbn = 978-0-534-24312-8
  | ref = CITEREFCasellaBerger2001
  | lccn = 2001025794
  | author2-link = Roger L. Berger
 }}

== External links ==
* [http://www.elektro-energetika.cz/calculations/ro.php?language=english Online calculator of Uniform distribution (continuous)]

{{ProbDistributions|continuous-bounded}}
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{{DEFAULTSORT:Uniform Distribution (Continuous)}}
[[Category:Continuous distributions]]
[[Category:Location-scale family probability distributions]]

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