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From: John M. <joh...@us...> - 2007-02-22 16:52:42
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Update of /cvsroot/boost-sandbox/boost-sandbox/libs/math_functions/doc/distributions In directory sc8-pr-cvs3.sourceforge.net:/tmp/cvs-serv28905/libs/math_functions/doc/distributions Modified Files: Tag: math_toolkit binomial.qbk non_members.qbk Log Message: A couple of minor using declaration adjustments. Updated docs: adjusted titles and added more information on the kurtosis. Index: non_members.qbk =================================================================== RCS file: /cvsroot/boost-sandbox/boost-sandbox/libs/math_functions/doc/distributions/Attic/non_members.qbk,v retrieving revision 1.1.2.8 retrieving revision 1.1.2.9 diff -u -d -r1.1.2.8 -r1.1.2.9 --- non_members.qbk 23 Dec 2006 13:35:45 -0000 1.1.2.8 +++ non_members.qbk 22 Feb 2007 16:52:36 -0000 1.1.2.9 @@ -291,17 +291,24 @@ Returns the 'proper' kurtosis (normalized fourth moment) of the distribution /dist/. +kertosis = [beta][sub 2][space]= [mu][sub 4][space] / [mu][sub 2][super 2] + +Where [mu][sub i][space] is the i'th central moment of the distribution, and +in particular [mu][sub 2][space] is the variance of the distribution. + +The kurtosis is a measure of the "peakedness" of a distribution. + Note that the literature definition of kurtosis is confusing. -The definition used here is that used by +The definition used here is that used by for example [@http://mathworld.wolfram.com/Kurtosis.html Wolfram MathWorld] (that includes a table of formulae for kurtosis excess for various distributions) but NOT the definition of [@http://en.wikipedia.org/wiki/Kurtosis kurtosis used by Wikipedia] -whose tables of distributions all show the (same) formulae labelled kurtosis. +which treats "kurtosis" and "kurtosis excess" as the same quantity. kurtosis_excess = 'proper' kurtosis - 3 -This subtraction of 3 is convenient so that the kurtosis excess +This subtraction of 3 is convenient so that the ['kurtosis excess] of a normal distribution is zero. This function may return a __domain_error if the distribution does not have @@ -316,6 +323,15 @@ Returns the kurtosis excess of the distribution /dist/. +kurtosis excess = [gamma][sub 2][space]= [mu][sub 4][space] / [mu][sub 2][super 2][space]- 3 = kurtosis - 3 + +Where [mu][sub i][space] is the i'th central moment of the distribution, and +in particular [mu][sub 2][space] is the variance of the distribution. + +The kurtosis excess is a measure of the "peakedness" of a distribution, and +is more widely used than the "kurtosis proper". It is defined so that +the kurtosis excess of a normal distribution is zero. + This function may return a __domain_error if the distribution does not have a defined kurtosis excess. Index: binomial.qbk =================================================================== RCS file: /cvsroot/boost-sandbox/boost-sandbox/libs/math_functions/doc/distributions/Attic/binomial.qbk,v retrieving revision 1.1.2.14 retrieving revision 1.1.2.15 diff -u -d -r1.1.2.14 -r1.1.2.15 --- binomial.qbk 4 Dec 2006 18:09:20 -0000 1.1.2.14 +++ binomial.qbk 22 Feb 2007 16:52:36 -0000 1.1.2.15 @@ -349,7 +349,7 @@ [[quantile][Since the cdf is non-linear in variate /k/ none of the inverse incomplete beta functions can be used here. Instead the quantile is found numerically using a derivative free method - ([link math_toolkit.toolkit.roots2 TOMS Algorithm 748]).]] + ([link math_toolkit.toolkit.internals1.roots2 TOMS Algorithm 748]).]] [[quantile from the complement][Found numerically as above.]] [[mean][ `p * n` ]] [[variance][ `p * n * (1-p)` ]] |