Uniform distribution
Story
Outcomes are restricted to a given continuous range and every outcome in that range has equal probability.
Example
Anything in which all possibilities are equally likely. This is, perhaps surprisingly, rarely encountered.
Parameters
The Uniform distribution is not defined on an infinite or semi-infinite domain, so finite lower and upper bounds, \(\alpha\) and \(\beta\), respectively, are necessary parameters.
Support
The Uniform distribution is supported on the interval \([\alpha, \beta]\).
Probability density function
\[\begin{split}\begin{align}
f(y;\alpha, \beta) = \left\{\begin{array}{ccl}\displaystyle{\frac{1}{\beta-\alpha}}&&\alpha\le y\le\beta\\[0.5em] 0 && \text{otherwise.}\end{array}\right.
\end{align}\end{split}\]
Cumulative distribution function
\[\begin{split}\begin{align}
F(y; \alpha, \beta) = \left\{\begin{array}{ccl} 0 && y < a \\[0.5em]\displaystyle{\frac{y-\alpha}{\beta-\alpha}}&&\alpha\le y\le\beta\\[0.5em] 1 && y > \beta\end{array}\right.
\end{align}\end{split}\]
Moments
Mean: \(\displaystyle{\frac{\alpha + \beta}{2}}\)
Variance: \(\displaystyle{\frac{(\beta - \alpha)^2}{12}}\)
Usage
Package |
Syntax |
|---|---|
NumPy |
|
SciPy |
|
Distributions.jl |
|
Stan |
|