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)` ]]
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