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Changed support range to avoid a discontinuity at zero.

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[SVN r42125]
This commit is contained in:
Paul A. Bristow
2007-12-17 15:59:20 +00:00
parent c327e0ac0f
commit 18d03d57f7
1 changed files with 20 additions and 3 deletions
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23
doc/sf_and_dist/dist_tutorial.qbk
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@@ -158,13 +158,32 @@ example, we would write:
`pdf(binomial_distribution<RealType>(n, p), k);`
The distribution (effectively the random variate) is said to be 'supported' over a range that is
The ranges of random variate values that are permitted and are supported can be
tested by using two functions `range` and `support`.
The distribution (effectively the random variate) is said to be 'supported'
over a range that is
[@http://en.wikipedia.org/wiki/Probability_distribution
"the smallest closed set whose complement has probability zero"].
MathWorld uses the word 'defined' for this range.
Non-mathematicians might say it means the 'interesting' smallest range
of random variate x that has the cdf going from zero to unity.
Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity.
For most distributions, with probability distribution functions one might describe
as 'well-behaved', we have decided that it is most useful for the supported range
to exclude random variate values like exact zero *if the end point is discontinuous*.
For example, the Weibull (scale 1, shape 1) distribution smoothly heads for unity
as the random variate x declines towards zero.
But at x = zero, the value of the pdf is suddenly exactly zero, by definition.
If you are plotting the PDF, or otherwise calculating,
zero is not the most useful value for the lower limit of supported, as we discovered.
So for this, and similar distributions,
we have decided it is most numerically useful to use
the closest value to zero, min_value, for the limit of the supported range.
(The `range` remains from zero, so you will still get `pdf(weibull, 0) == 0`).
(Exponential and gamma distributions have similarly discontinuous functions).
Mathematically, the functions may make sense with an (+ or -) infinite value,
but except for a few special cases (in the Normal and Cauchy distributions)
this implementation limits random variates to finite values from the `max`
@@ -172,8 +191,6 @@ to `min` for the `RealType`.
(See [link math_toolkit.backgrounders.implementation.handling_of_floating_point_infinity
Handling of Floating-Point Infinity] for rationale).
The range of random variate values that is permitted and supported can be
tested by using two functions `range` and `support`.
[note