mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-10-31 04:50:34 -04:00
85 lines
3.7 KiB
Plaintext
85 lines
3.7 KiB
Plaintext
[section:variates Random Variates and Distribution Parameters]
|
|
|
|
[@http://en.wikipedia.org/wiki/Random_variate Random variates]
|
|
and [@http://en.wikipedia.org/wiki/Parameter distribution parameters]
|
|
are conventionally distinguished (for example in Wikipedia and Wolfram MathWorld
|
|
by placing a semi-colon after the __random_variate (whose value you 'choose'),
|
|
to separate the variate from the parameter(s) that defines the shape of the distribution.
|
|
|
|
For example, the binomial distribution has two parameters:
|
|
n (the number of trials) and p (the probability of success on one trial).
|
|
It also has the __random_variate /k/: the number of successes observed.
|
|
This means the probability density\/mass function (pdf) is written as ['f(k; n, p)].
|
|
|
|
Translating this into code the `binomial_distribution` constructor
|
|
therefore has two parameters:
|
|
|
|
binomial_distribution(RealType n, RealType p);
|
|
|
|
While the function `pdf` has one argument specifying the distribution type
|
|
(which includes its parameters, if any),
|
|
and a second argument for the __random_variate. So taking our binomial distribution
|
|
example, we would write:
|
|
|
|
pdf(binomial_distribution<RealType>(n, p), k);
|
|
|
|
[endsect]
|
|
|
|
[section:dist_params Discrete Probability Distributions]
|
|
|
|
Note that the [@http://en.wikipedia.org/wiki/Discrete_probability_distribution
|
|
discrete distributions], including the binomial, negative binomial, Poisson & Bernoulli,
|
|
are all mathematically defined as discrete functions:
|
|
only integral values of the __random_variate are envisaged
|
|
and the functions are only defined at these integral values.
|
|
However because the method of calculation often uses continuous functions,
|
|
it is convenient to treat them as if they were continuous functions,
|
|
and permit non-integral values of their parameters.
|
|
|
|
To enforce a strict mathematical model,
|
|
users may use floor or ceil functions on the __random_variate,
|
|
prior to calling the distribution function, to enforce integral values.
|
|
|
|
For similar reasons, in continuous distributions, parameters like degrees of freedom
|
|
that might appear to be integral, are treated as real values
|
|
(and are promoted from integer to floating-point if necessary).
|
|
In this case however, that there are a small number of situations where non-integral
|
|
degrees of freedom do have a genuine meaning.
|
|
|
|
Generally speaking there is no loss of performance from allowing real-values
|
|
parameters: the underlying special functions contain optimizations for
|
|
integer-valued arguments when applicable.
|
|
|
|
[caution
|
|
The quantile function of a discrete distribution will by
|
|
default return an integer result that has been
|
|
/rounded outwards/. That is to say lower quantiles (where the probability is
|
|
less than 0.5) are rounded downward, and upper quantiles (where the probability
|
|
is greater than 0.5) are rounded upwards. This behaviour
|
|
ensures that if an X% quantile is requested, then /at least/ the requested
|
|
coverage will be present in the central region, and /no more than/
|
|
the requested coverage will be present in the tails.
|
|
|
|
This behaviour can be changed so that the quantile functions are rounded
|
|
differently, or even return a real-valued result using
|
|
[link math_toolkit.pol_overview Policies]. It is strongly
|
|
recommended that you read the tutorial
|
|
[link math_toolkit.pol_tutorial.understand_dis_quant
|
|
Understanding Quantiles of Discrete Distributions] before
|
|
using the quantile function on a discrete distribution. The
|
|
[link math_toolkit.pol_ref.discrete_quant_ref reference docs]
|
|
describe how to change the rounding policy
|
|
for these distributions.
|
|
]
|
|
|
|
[endsect]
|
|
|
|
[/
|
|
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).
|
|
]
|
|
|
|
|