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- [section:uniform_dist Uniform Distribution]
- ``#include <boost/math/distributions/uniform.hpp>``
- namespace boost{ namespace math{
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- class uniform_distribution;
-
- typedef uniform_distribution<> uniform;
- template <class RealType, class ``__Policy``>
- class uniform_distribution
- {
- public:
- typedef RealType value_type;
- uniform_distribution(RealType lower = 0, RealType upper = 1); // Constructor.
- : m_lower(lower), m_upper(upper) // Default is standard uniform distribution.
- // Accessor functions.
- RealType lower()const;
- RealType upper()const;
- }; // class uniform_distribution
-
- }} // namespaces
-
- The uniform distribution, also known as a rectangular distribution,
- is a probability distribution that has constant probability.
- The [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 continuous uniform distribution]
- is a distribution with the
- [@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
- [expression f(x) =1 / (upper - lower) [sixemspace] for lower < x < upper]
- [expression f(x) =zero [sixemspace] for x < lower or x > upper]
-
- and in this implementation:
-
- [expression 1 / (upper - lower) [sixemspace] for x = lower or x = upper]
- The choice of /x = lower/ or /x = upper/ is made because statistical use of this distribution judged is most likely:
- the method of maximum likelihood uses this definition.
- There is also a [@http://en.wikipedia.org/wiki/Discrete_uniform_distribution *discrete* uniform distribution].
- Parameters lower and upper can be any finite value.
- The [@http://en.wikipedia.org/wiki/Random_variate random variate]
- /x/ must also be finite, and is supported /lower <= x <= upper/.
- The lower parameter is also called the
- [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm location parameter],
- [@http://en.wikipedia.org/wiki/Location_parameter that is where the origin of a plot will lie],
- and (upper - lower) is also called the [@http://en.wikipedia.org/wiki/Scale_parameter scale parameter].
- The following graph illustrates how the
- [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
- varies with the shape parameter:
- [graph uniform_pdf]
- Likewise for the CDF:
- [graph uniform_cdf]
- [h4 Member Functions]
- uniform_distribution(RealType lower = 0, RealType upper = 1);
-
- Constructs a [@http://en.wikipedia.org/wiki/uniform_distribution
- uniform distribution] with lower /lower/ (a) and upper /upper/ (b).
- Requires that the /lower/ and /upper/ parameters are both finite;
- otherwise if infinity or NaN then calls __domain_error.
- RealType lower()const;
-
- Returns the /lower/ parameter of this distribution.
-
- RealType upper()const;
-
- Returns the /upper/ parameter of this distribution.
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
- that are generic to all distributions are supported: __usual_accessors.
- The domain of the random variable is any finite value,
- but the supported range is only /lower/ <= x <= /upper/.
- [h4 Accuracy]
- The uniform distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two.
- [h4 Implementation]
- In the following table a is the /lower/ parameter of the distribution,
- b is the /upper/ parameter,
- /x/ is the random variate, /p/ is the probability and /q = 1-p/.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using the relation: pdf = 0 for x < a, 1 / (b - a) for a <= x <= b, 0 for x > b ]]
- [[cdf][Using the relation: cdf = 0 for x < a, (x - a) / (b - a) for a <= x <= b, 1 for x > b]]
- [[cdf complement][Using the relation: q = 1 - p, (b - x) / (b - a) ]]
- [[quantile][Using the relation: x = p * (b - a) + a; ]]
- [[quantile from the complement][x = -q * (b - a) + b ]]
- [[mean][(a + b) / 2 ]]
- [[variance][(b - a) [super 2] / 12 ]]
- [[mode][any value in \[a, b\] but a is chosen. (Would NaN be better?) ]]
- [[skewness][0]]
- [[kurtosis excess][-6/5 = -1.2 exactly. (kurtosis - 3)]]
- [[kurtosis][9/5]]
- ]
- [h4 References]
- * [@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 Wikpedia continuous uniform distribution]
- * [@http://mathworld.wolfram.com/UniformDistribution.html Weisstein, Weisstein, Eric W. "Uniform Distribution." From MathWorld--A Wolfram Web Resource.]
- * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm]
- [endsect] [/section:uniform_dist Uniform]
- [/
- Copyright 2006 John Maddock and Paul A. Bristow.
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt).
- ]
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