# Boost-Commit :

Subject: [Boost-commit] svn:boost r65695 - trunk/libs/math/doc/sf_and_dist/distributions
From: pbristow_at_[hidden]
Date: 2010-09-30 14:03:34

Author: pbristow
Date: 2010-09-30 14:03:33 EDT (Thu, 30 Sep 2010)
New Revision: 65695
URL: http://svn.boost.org/trac/boost/changeset/65695

Log:
Corrected formulae for pdf and cdf.
Text files modified:
trunk/libs/math/doc/sf_and_dist/distributions/inverse_chi_squared.qbk | 17 +++++++++++++----
1 files changed, 13 insertions(+), 4 deletions(-)

Modified: trunk/libs/math/doc/sf_and_dist/distributions/inverse_chi_squared.qbk
==============================================================================
--- trunk/libs/math/doc/sf_and_dist/distributions/inverse_chi_squared.qbk (original)
+++ trunk/libs/math/doc/sf_and_dist/distributions/inverse_chi_squared.qbk 2010-09-30 14:03:33 EDT (Thu, 30 Sep 2010)
@@ -57,7 +57,7 @@

-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 @@

or you can write `inverse_chi_squared my_invchisqr(2, 3);`]

+For degrees of freedom parameter [nu],
+the (*unscaled*) inverse chi_squared distribution is defined by the probability density function (PDF):
+
+__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]) = [Gamma]([nu]/2, 1/2x) / [Gamma]([nu]/2)
+
For degrees of freedom parameter [nu] and scale parameter [xi],
-it is defined by the probability density function (PDF):
+the *scaled* 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], [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)
+__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]: