EQ2EMu/EQ2/source/depends/boost_1_72_0/libs/math/doc/distributions/uniform.qbk

[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).
]