From 669ce3a713437d76dc0da87ab4b1cf69d6417baa Mon Sep 17 00:00:00 2001 From: "Paul A. Bristow" Date: Thu, 30 Sep 2010 18:03:33 +0000 Subject: [PATCH] Corrected formulae for pdf and cdf. [SVN r65695] --- .../distributions/inverse_chi_squared.qbk | 19 ++++++++++++++----- 1 file changed, 14 insertions(+), 5 deletions(-) diff --git a/doc/sf_and_dist/distributions/inverse_chi_squared.qbk b/doc/sf_and_dist/distributions/inverse_chi_squared.qbk index ed82b1a97..943db4295 100644 --- a/doc/sf_and_dist/distributions/inverse_chi_squared.qbk +++ b/doc/sf_and_dist/distributions/inverse_chi_squared.qbk @@ -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]: