In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.[1] The bounds are defined by the parameters,
and
which are the minimum and maximum values. The interval can either be closed (i.e.
) or open (i.e.
).[2] Therefore, the distribution is often abbreviated
where
stands for uniform distribution.[1] The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable
under no constraint other than that it is contained in the distribution's support.[3]
Quick Facts Notation, Parameters ...
Continuous uniform
Probability density function Using maximum convention |
Cumulative distribution function |
Notation |
![{\displaystyle {\mathcal {U}}_{[a,b]}}](//wikimedia.org/api/rest_v1/media/math/render/svg/906b38f0905adef68e3c8c7ca6de15858f7742ce) |
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Parameters |
![{\displaystyle -\infty <a<b<\infty }](//wikimedia.org/api/rest_v1/media/math/render/svg/b96ee7d3634294ef2eef00ffacad47efe5179d97) |
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Support |
![{\displaystyle [a,b]}](//wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935) |
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PDF |
![{\displaystyle {\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/648692e002b720347c6c981aeec2a8cca7f4182f) |
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CDF |
![{\displaystyle {\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/2948c023c98e2478806980eb7f5a03810347a568) |
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Mean |
![{\displaystyle {\tfrac {1}{2}}(a+b)}](//wikimedia.org/api/rest_v1/media/math/render/svg/83f8e71092f95652ba4e65a6916c144aa470f4ec) |
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Median |
![{\displaystyle {\tfrac {1}{2}}(a+b)}](//wikimedia.org/api/rest_v1/media/math/render/svg/83f8e71092f95652ba4e65a6916c144aa470f4ec) |
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Mode |
![{\displaystyle {\text{any value in }}(a,b)}](//wikimedia.org/api/rest_v1/media/math/render/svg/0ea276bb6b04c5931ccd95309c3c2e6771e08e5b) |
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Variance |
![{\displaystyle {\tfrac {1}{12}}(b-a)^{2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/95f6f2aef440271aa37dec67fe279bb74e4398a4) |
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MAD |
![{\displaystyle {\tfrac {1}{4}}(b-a)}](//wikimedia.org/api/rest_v1/media/math/render/svg/3c0cc468ccb04048d89311be7fba111d4fa8777c) |
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Skewness |
![{\displaystyle 0}](//wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
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Excess kurtosis |
![{\displaystyle -{\tfrac {6}{5}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/75b6c73a703b1145c67260493067b32d5879aabf) |
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Entropy |
![{\displaystyle \log(b-a)}](//wikimedia.org/api/rest_v1/media/math/render/svg/9f2371b977ce272d6f71920e6f240d1df095f58a) |
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MGF |
![{\displaystyle {\begin{cases}{\frac {\mathrm {e} ^{tb}-\mathrm {e} ^{ta}}{t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0d62c976b0480469b17966acdb27fb9b1fa5066d) |
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CF |
![{\displaystyle {\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/33f609d1d6670ceb18125fd65486b73be354c22f) |
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