mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-11-03 21:40:52 -05:00
135 lines
4.6 KiB
Plaintext
135 lines
4.6 KiB
Plaintext
[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]:
|
|
|
|
f(x) =
|
|
|
|
* 1 / (upper - lower) for lower < x < upper
|
|
|
|
* zero for x < lower or x > upper
|
|
|
|
and in this implementation:
|
|
|
|
* 1 / (upper - lower) 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).
|
|
]
|
|
|