Continuous uniform distribution information

Continuous uniform
	Probability density function; Using maximum convention
	Cumulative distribution function
Notation
Parameters
Support
PDF
CDF
Mean
Median
Mode
Variance
MAD
Skewness
Excess kurtosis
Entropy
MGF
CF

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, $a$ and $b,$ which are the minimum and maximum values. The interval can either be closed (i.e. $[a,b]$ ) or open (i.e. $(a,b)$ ).^[2] Therefore, the distribution is often abbreviated $U(a,b),$ where $U$ 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 $X$ under no constraint other than that it is contained in the distribution's support.^[3]

^ ^a ^b Dekking, Michel (2005). A modern introduction to probability and statistics : understanding why and how. London, UK: Springer. pp. 60–61. ISBN 978-1-85233-896-1.
^ Walpole, Ronald; et al. (2012). Probability & Statistics for Engineers and Scientists. Boston, USA: Prentice Hall. pp. 171–172. ISBN 978-0-321-62911-1.
^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics. 150 (2): 219–230. CiteSeerX 10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014.

[:0-1] Dekking, Michel (2005). A modern introduction to probability and statistics : understanding why and how. London, UK: Springer. pp. 60–61. ISBN 978-1-85233-896-1.

[:3-2] Walpole, Ronald; et al. (2012). Probability & Statistics for Engineers and Scientists. Boston, USA: Prentice Hall. pp. 171–172. ISBN 978-0-321-62911-1.

[3] Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model". Journal of Econometrics. 150 (2): 219–230. CiteSeerX 10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014.

Continuous uniform distribution information

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Continuous uniform distribution

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Exponential distribution

Probability density function

Symmetric probability distribution

Generalized Pareto distribution

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Generalized normal distribution

Weibull distribution

Probability density function Using maximum convention
Cumulative distribution function
Notation	${\mathcal {U}}_{[a,b]}$
Parameters	$-\infty <a<b<\infty$
Support	$[a,b]$
PDF	${\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}$
CDF	${\begin{cases}0&{\text{for }}x<a\\{\frac {x-a}{b-a}}&{\text{for }}x\in [a,b]\\1&{\text{for }}x>b\end{cases}}$
Mean	${\tfrac {1}{2}}(a+b)$
Median	${\tfrac {1}{2}}(a+b)$
Mode	${\text{any value in }}(a,b)$
Variance	${\tfrac {1}{12}}(b-a)^{2}$
MAD	${\tfrac {1}{4}}(b-a)$
Skewness	$0$
Excess kurtosis	$-{\tfrac {6}{5}}$
Entropy	$\ln(b-a)$
MGF	${\begin{cases}{\frac {\mathrm {e} ^{tb}-\mathrm {e} ^{ta}}{t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}$
CF	${\begin{cases}{\frac {\mathrm {e} ^{\mathrm {i} tb}-\mathrm {e} ^{\mathrm {i} ta}}{\mathrm {i} t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}$