In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the continuous uniform distribution or rectangular distribution is a family of
symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.
The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
(e.g.
, b or
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* Open (Blues Image album), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
(e.g. (a, b)).
Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution.
The difference between the bounds defines the interval length; all
intervals of the same length on the distribution's
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
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.
Definitions
Probability density function
The
probability density function of the continuous uniform distribution is:
:
The values of ''f''(''x'') at the two boundaries ''a'' and ''b'' are usually unimportant because they do not alter the values of the integrals of over any interval, nor of or any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be . 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 ''f''(''a'') or ''f''(''b'') to be , since then the inverse transform of many
integral transforms 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. Also, it is consistent with the
sign function which has no such ambiguity.
Graphically, the
probability density function is portrayed as a rectangle where is the base and is the height. As the distance between a and b increases, the density at any particular value within the distribution boundaries decreases.
Since the
probability density function integrates to 1, the height of the probability density function decreases as the base length increases.
In terms of mean ''μ'' and variance ''σ''
2, the probability density may be written as:
:
Cumulative distribution function
The
cumulative distribution function is:
:
Its inverse is:
: