| [section:uniform_dist Uniform Distribution] |
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| ``#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]: |
| |
| f(x) = |
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| * 1 / (upper - lower) for lower < x < upper |
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| * zero for x < lower or x > upper |
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| and in this implementation: |
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| * 1 / (upper - lower) for x = lower or x = upper |
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| 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. |
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| There is also a [@http://en.wikipedia.org/wiki/Discrete_uniform_distribution *discrete* uniform distribution]. |
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| Parameters lower and upper can be any finite value. |
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| 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]. |
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| The following graph illustrates how the |
| [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF] |
| varies with the shape parameter: |
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| [graph uniform_pdf] |
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| Likewise for the CDF: |
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| [graph uniform_cdf] |
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| [h4 Member Functions] |
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| 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. |
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| RealType lower()const; |
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| Returns the /lower/ parameter of this distribution. |
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| RealType upper()const; |
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| Returns the /upper/ parameter of this distribution. |
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| [h4 Non-member Accessors] |
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| All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] |
| that are generic to all distributions are supported: __usual_accessors. |
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| The domain of the random variable is any finite value, |
| but the supported range is only /lower/ <= x <= /upper/. |
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| [h4 Accuracy] |
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| The uniform distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two. |
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| [h4 Implementation] |
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| 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] |
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| [endsect][/section:uniform_dist Uniform] |
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| [/ |
| 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). |
| ] |
| |