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Corrected formulae for pdf and cdf.

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[SVN r65695]
This commit is contained in:
Paul A. Bristow
2010-09-30 18:03:33 +00:00
parent bbc5edfe1c
commit 669ce3a713
1 changed files with 14 additions and 5 deletions
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19
doc/sf_and_dist/distributions/inverse_chi_squared.qbk
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@@ -57,7 +57,7 @@ See [@http://en.wikipedia.org/wiki/Conjugate_prior conjugate priors including a
See also __inverse_gamma_distrib and __chi_squared_distrib.
The inverse_chi_squared distribution is a psecial case of a inverse_gamma distribution
The inverse_chi_squared distribution is a special case of a inverse_gamma distribution
with [nu] (degrees_of_freedom) shape ([alpha]) and scale ([beta]) where
__spaces [alpha]= [nu] /2 and [beta] = [frac12].
@@ -72,14 +72,23 @@ If you want a `double` precision inverse_chi_squared distribution you can use
or you can write `inverse_chi_squared my_invchisqr(2, 3);`]
For degrees of freedom parameter [nu] and scale parameter [xi],
it is defined by the probability density function (PDF):
For degrees of freedom parameter [nu],
the (*unscaled*) inverse chi_squared distribution is defined by the probability density function (PDF):
__spaces f(x;[nu], [xi]) = 2 [super -[nu]/2] e[super(-1/2x)] x [super(-1-[nu]/2)] / [Gamma]([nu]/2)
__spaces f(x;[nu]) = 2[super -[nu]/2] x[super -[nu]/2-1] e[super -1/2x] / [Gamma]([nu]/2)
and Cumulative Density Function (CDF)
__spaces F(x;[nu], [xi]) = [Gamma]( [nu] /2, [nu][xi]/2x) / [Gamma]([nu] /2)
__spaces F(x;[nu]) = [Gamma]([nu]/2, 1/2x) / [Gamma]([nu]/2)
For degrees of freedom parameter [nu] and scale parameter [xi],
the *scaled* inverse chi_squared distribution is defined by the probability density function (PDF):
__spaces f(x;[nu], [xi]) = ([xi][nu]/2)[super [nu]/2] e[super -[nu][xi]/2x] x[super -1-[nu]/2] / [Gamma]([nu]/2)
and Cumulative Density Function (CDF)
__spaces F(x;[nu], [xi]) = [Gamma]([nu]/2, [nu][xi]/2x) / [Gamma]([nu]/2)
The following graphs illustrate how the PDF and CDF of the inverse chi_squared distribution
varies for a few values of parameters [nu] and [xi]: