Uniform Distribution

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##### Uniform Distribution

    #include <boost/math/distributions/uniform.hpp>

    namespace boost{ namespace math{
     template <class RealType = double,
               class Policy   = policies::policy<> >
     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 continuous uniform distribution is a distribution with the 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 discrete uniform distribution.

Parameters lower and upper can be any finite value.

The random variate x must also be finite, and is supported lower <= x <= upper.

The lower parameter is also called the location parameter, that is where the origin of a plot will lie, and (upper - lower) is also called the scale parameter.

The following graph illustrates how the probability density function PDF varies with the shape parameter:

Likewise for the CDF:

##### Member Functions

    uniform_distribution(RealType lower = 0, RealType upper = 1);

Constructs a 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.

##### Non-member Accessors

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

The domain of the random variable is any finite value, but the supported range is only lower <= x <= upper.

##### Accuracy

The uniform distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two.

##### 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.

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) 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

##### References

- Wikpedia continuous uniform distribution
- Weisstein, Weisstein, Eric W. "Uniform Distribution." From MathWorld--A Wolfram Web Resource.
- http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm