boost_1_45_0/libs/math/doc/sf_and_dist/distributions/non_members.qbk - nest-learning-thermostat/5.0/boost - Git at Google

 [section:nmp Non-Member Properties]

 Properties that are common to all distributions are accessed via non-member
 getter functions: non-membership allows more of these functions to be added over time,
 as the need arises.  Unfortunately the literature uses many different and
 confusing names to refer to a rather small number of actual concepts; refer
 to the [link concept_index concept index] to find the property you
 want by the name you are most familiar with.
 Or use the [link function_index function index]
 to go straight to the function you want if you already know its name.

 [h4 [#function_index]Function Index]

 * [link math.dist.cdf cdf].
 * [link math.dist.ccdf cdf complement].
 * [link math.dist.chf chf].
 * [link math.dist.hazard hazard].
 * __kurtosis.
 * __kurtosis_excess
 * __mean.
 * [link math.dist.median median].
 * __mode.
 * [link math.dist.pdf pdf].
 * [link math.dist.range range].
 * [link math.dist.quantile quantile].
 * [link math.dist.quantile_c quantile from the complement].
 * __skewness.
 * [link math.dist.sd standard_deviation].
 * [link math.dist.support support].
 * __variance.

 [h4 [#concept_index]Conceptual Index]

 * __ccdf.
 * __cdf.
 * __chf.
 * [link cdf_inv Inverse Cumulative Distribution Function].
 * [link survival_inv Inverse Survival Function].
 * __hazard
 * [link lower_critical Lower Critical Value].
 * __kurtosis.
 * __kurtosis_excess
 * __mean.
 * [link math.dist.median median].
 * __mode.
 * [link cdfPQ P].
 * [link percent Percent Point Function].
 * __pdf.
 * [link pmf Probability Mass Function].
 * [link math.dist.range range].
 * [link cdfPQ Q].
 * __quantile.
 * [link math.dist.quantile_c Quantile from the complement of the probability].
 * __skewness.
 * __sd
 * [link survival Survival Function].
 * [link math.dist.support support].
 * [link upper_critical Upper Critical Value].
 * __variance.

 [h4 [#math.dist.cdf]Cumulative Distribution Function]

    template <class RealType, class ``__Policy``>
    RealType cdf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

 The __cdf is the probability that
 the variable takes a value less than or equal to x.  It is equivalent
 to the integral from -infinity to x of the __pdf.

 This function may return a __domain_error if the random variable is outside
 the defined range for the distribution.

 For example, the following graph shows the cdf for the
 normal distribution:

 [$../graphs/cdf.png]

 [h4 [#math.dist.ccdf]Complement of the Cumulative Distribution Function]

    template <class Distribution, class RealType>
    RealType cdf(const ``['Unspecified-Complement-Type]``<Distribution, RealType>& comp);

 The complement of the __cdf
 is the probability that
 the variable takes a value greater than x.  It is equivalent
 to the integral from x to infinity of the __pdf, or 1 minus the __cdf of x.

 This is also known as the survival function.

 This function may return a __domain_error if the random variable is outside
 the defined range for the distribution.

 In this library, it is obtained by wrapping the arguments to the `cdf`
 function in a call to `complement`, for example:

    // standard normal distribution object:
    boost::math::normal norm;
    // print survival function for x=2.0:
    std::cout << cdf(complement(norm, 2.0)) << std::endl;

 For example, the following graph shows the __complement of the cdf for the
 normal distribution:

 [$../graphs/survival.png]

 See __why_complements for why the complement is useful and when it should be used.

 [h4 [#math.dist.hazard]Hazard Function]

    template <class RealType, class ``__Policy``>
    RealType hazard(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

 Returns the __hazard of /x/ and distibution /dist/.

 This function may return a __domain_error if the random variable is outside
 the defined range for the distribution.

 [equation hazard]

 [caution
 Some authors refer to this as the conditional failure
 density function rather than the hazard function.]

 [h4 [#math.dist.chf]Cumulative Hazard Function]

    template <class RealType, class ``__Policy``>
    RealType chf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

 Returns the __chf of /x/ and distibution /dist/.

 This function may return a __domain_error if the random variable is outside
 the defined range for the distribution.

 [equation chf]

 [caution
 Some authors refer to this as simply the "Hazard Function".]

 [h4 [#math.dist.mean]mean]

    template<class RealType, class ``__Policy``>
    RealType mean(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the mean of the distribution /dist/.

 This function may return a __domain_error if the distribution does not have
 a defined mean (for example the Cauchy distribution).

 [h4 [#math.dist.median]median]

    template<class RealType, class ``__Policy``>
    RealType median(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the median of the distribution /dist/.

 [h4 [#math.dist.mode]mode]

    template<class RealType, ``__Policy``>
    RealType mode(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the mode of the distribution /dist/.

 This function may return a __domain_error if the distribution does not have
 a defined mode.

 [h4 [#math.dist.pdf]Probability Density Function]

    template <class RealType, class ``__Policy``>
    RealType pdf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

 For a continuous function, the probability density function (pdf) returns
 the probability that the variate has the value x.
 Since for continuous distributions the probability at a single point is actually zero,
 the probability is better expressed as the integral of the pdf between two points:
 see the __cdf.

 For a discrete distribution, the pdf is the probability that the
 variate takes the value x.

 This function may return a __domain_error if the random variable is outside
 the defined range for the distribution.

 For example, for a standard normal distribution the pdf looks like this:

 [$../graphs/pdf.png]

 [h4 [#math.dist.range]Range]

    template<class RealType, class ``__Policy``>
    std::pair<RealType, RealType> range(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the valid range of the random variable over distribution /dist/.

 [h4 [#math.dist.quantile]Quantile]

    template <class RealType, class ``__Policy``>
    RealType quantile(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& p);

 The quantile is best viewed as the inverse of the __cdf, it returns
 a value /x/ such that `cdf(dist, x) == p`.

 This is also known as the /percent point function/, or a /percentile/, it is
 also the same as calculating the ['lower critical value] of a distribution.

 This function returns a __domain_error if the probability lies outside [0,1].
 The function may return an __overflow_error if there is no finite value
 that has the specified probability.

 The following graph shows the quantile function for a standard normal
 distribution:

 [$../graphs/quantile.png]

 [h4 [#math.dist.quantile_c]Quantile from the complement of the probability.]
 [link complements complements]


    template <class Distribution, class RealType>
    RealType quantile(const ``['Unspecified-Complement-Type]``<Distribution, RealType>& comp);

 This is the inverse of the __ccdf.  It is calculated by wrapping
 the arguments in a call to the quantile function in a call to
 /complement/.  For example:

    // define a standard normal distribution:
    boost::math::normal norm;
    // print the value of x for which the complement
    // of the probability is 0.05:
    std::cout << quantile(complement(norm, 0.05)) << std::endl;

 The function computes a value /x/ such that
 `cdf(complement(dist, x)) == q` where /q/ is complement of the
 probability.

 [link why_complements Why complements?]

 This function is also called the inverse survival function, and is the
 same as calculating the ['upper critical value] of a distribution.

 This function returns a __domain_error if the probablity lies outside [0,1].
 The function may return an __overflow_error if there is no finite value
 that has the specified probability.

 The following graph show the inverse survival function for the normal
 distribution:

 [$../graphs/survival_inv.png]

 [h4 [#math.dist.sd]Standard Deviation]

    template <class RealType, class ``__Policy``>
    RealType standard_deviation(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the standard deviation of distribution /dist/.

 This function may return a __domain_error if the distribution does not have
 a defined standard deviation.

 [h4 [#math.dist.support]support]

    template<class RealType, class ``__Policy``>
    std::pair<RealType, RealType> support(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the supported range of random variable over the distribution /dist/.

 The distribution is said to be 'supported' over a range that is
 [@http://en.wikipedia.org/wiki/Probability_distribution
  "the smallest closed set whose complement has probability zero"].
 Non-mathematicians might say it means the 'interesting' smallest range
 of random variate x that has the cdf going from zero to unity.
 Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity.

 [h4 [#math.dist.variance]Variance]

    template <class RealType, class ``__Policy``>
    RealType variance(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the variance of the distribution /dist/.

 This function may return a __domain_error if the distribution does not have
 a defined variance.

 [h4 [#math.dist.skewness]Skewness]

    template <class RealType, class ``__Policy``>
    RealType skewness(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 Returns the skewness of the distribution /dist/.

 This function may return a __domain_error if the distribution does not have
 a defined skewness.

 [h4 [#math.dist.kurtosis]Kurtosis]

    template <class RealType, class ``__Policy``>
    RealType kurtosis(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 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 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]
 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]
 of a normal distribution is zero.

 This function may return a __domain_error if the distribution does not have
 a defined kurtosis.

 'Proper' kurtosis can have a value from zero to + infinity.

 [h4 [#math.dist.kurtosis_excess]Kurtosis excess]

    template <class RealType, ``__Policy``>
    RealType kurtosis_excess(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

 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.

 Kurtosis excess can have a value from -2 to + infinity.

   kurtosis = kurtosis_excess +3;

 The kurtosis excess of a normal distribution is zero.

 [h4 [#cdfPQ]P and Q]

 The terms P and Q are sometimes used to refer to the __cdf
 and its [link math.dist.ccdf complement] respectively.
 Lowercase p and q are sometimes used to refer to the values returned
 by these functions.

 [h4 [#percent]Percent Point Function]

 The percent point function, also known as the percentile, is the same as
 the __quantile.

 [h4 [#cdf_inv]Inverse CDF Function.]

 The inverse of the cumulative distribution function, is the same as the
 __quantile.

 [h4 [#survival_inv]Inverse Survival Function.]

 The inverse of the survival function, is the same as computing the
 [link math.dist.quantile_c quantile
 from the complement of the probability].

 [h4 [#pmf]Probability Mass Function]

 The Probability Mass Function is the same as the __pdf.

 The term Mass Function is usually applied to discrete distributions,
 while the term __pdf applies to continuous distributions.

 [h4 [#lower_critical]Lower Critical Value.]

 The lower critical value calculates the value of the random variable
 given the area under the left tail of the distribution.
 It is equivalent to calculating the __quantile.

 [h4 [#upper_critical]Upper Critical Value.]

 The upper critical value calculates the value of the random variable
 given the area under the right tail of the distribution.  It is equivalent to
 calculating the [link math.dist.quantile_c quantile from the complement of the
 probability].

 [h4 [#survival]Survival Function]

 Refer to the __ccdf.

 [endsect][/section:nmp Non-Member Properties]


 [/ non_members.qbk
   Copyright 2006 John Maddock and Paul A. Bristow.
   Distributed under the Boost Software License, Version 1.0.
   (See accompanying file LICENSE_1_0.txt or copy at
   http://www.boost.org/LICENSE_1_0.txt).
 ]
	[section:nmp Non-Member Properties]

	Properties that are common to all distributions are accessed via non-member
	getter functions: non-membership allows more of these functions to be added over time,
	as the need arises. Unfortunately the literature uses many different and
	confusing names to refer to a rather small number of actual concepts; refer
	to the [link concept_index concept index] to find the property you
	want by the name you are most familiar with.
	Or use the [link function_index function index]
	to go straight to the function you want if you already know its name.

	[h4 [#function_index]Function Index]

	* [link math.dist.cdf cdf].
	* [link math.dist.ccdf cdf complement].
	* [link math.dist.chf chf].
	* [link math.dist.hazard hazard].
	* __kurtosis.
	* __kurtosis_excess
	* __mean.
	* [link math.dist.median median].
	* __mode.
	* [link math.dist.pdf pdf].
	* [link math.dist.range range].
	* [link math.dist.quantile quantile].
	* [link math.dist.quantile_c quantile from the complement].
	* __skewness.
	* [link math.dist.sd standard_deviation].
	* [link math.dist.support support].
	* __variance.

	[h4 [#concept_index]Conceptual Index]

	* __ccdf.
	* __cdf.
	* __chf.
	* [link cdf_inv Inverse Cumulative Distribution Function].
	* [link survival_inv Inverse Survival Function].
	* __hazard
	* [link lower_critical Lower Critical Value].
	* __kurtosis.
	* __kurtosis_excess
	* __mean.
	* [link math.dist.median median].
	* __mode.
	* [link cdfPQ P].
	* [link percent Percent Point Function].
	* __pdf.
	* [link pmf Probability Mass Function].
	* [link math.dist.range range].
	* [link cdfPQ Q].
	* __quantile.
	* [link math.dist.quantile_c Quantile from the complement of the probability].
	* __skewness.
	* __sd
	* [link survival Survival Function].
	* [link math.dist.support support].
	* [link upper_critical Upper Critical Value].
	* __variance.

	[h4 [#math.dist.cdf]Cumulative Distribution Function]

	template <class RealType, class ``__Policy``>
	RealType cdf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

	The __cdf is the probability that
	the variable takes a value less than or equal to x. It is equivalent
	to the integral from -infinity to x of the __pdf.

	This function may return a __domain_error if the random variable is outside
	the defined range for the distribution.

	For example, the following graph shows the cdf for the
	normal distribution:

	[$../graphs/cdf.png]

	[h4 [#math.dist.ccdf]Complement of the Cumulative Distribution Function]

	template <class Distribution, class RealType>
	RealType cdf(const ``['Unspecified-Complement-Type]``<Distribution, RealType>& comp);

	The complement of the __cdf
	is the probability that
	the variable takes a value greater than x. It is equivalent
	to the integral from x to infinity of the __pdf, or 1 minus the __cdf of x.

	This is also known as the survival function.

	This function may return a __domain_error if the random variable is outside
	the defined range for the distribution.

	In this library, it is obtained by wrapping the arguments to the `cdf`
	function in a call to `complement`, for example:

	// standard normal distribution object:
	boost::math::normal norm;
	// print survival function for x=2.0:
	std::cout << cdf(complement(norm, 2.0)) << std::endl;

	For example, the following graph shows the __complement of the cdf for the
	normal distribution:

	[$../graphs/survival.png]

	See __why_complements for why the complement is useful and when it should be used.

	[h4 [#math.dist.hazard]Hazard Function]

	template <class RealType, class ``__Policy``>
	RealType hazard(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

	Returns the __hazard of /x/ and distibution /dist/.

	This function may return a __domain_error if the random variable is outside
	the defined range for the distribution.

	[equation hazard]

	[caution
	Some authors refer to this as the conditional failure
	density function rather than the hazard function.]

	[h4 [#math.dist.chf]Cumulative Hazard Function]

	template <class RealType, class ``__Policy``>
	RealType chf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

	Returns the __chf of /x/ and distibution /dist/.

	This function may return a __domain_error if the random variable is outside
	the defined range for the distribution.

	[equation chf]

	[caution
	Some authors refer to this as simply the "Hazard Function".]

	[h4 [#math.dist.mean]mean]

	template<class RealType, class ``__Policy``>
	RealType mean(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the mean of the distribution /dist/.

	This function may return a __domain_error if the distribution does not have
	a defined mean (for example the Cauchy distribution).

	[h4 [#math.dist.median]median]

	template<class RealType, class ``__Policy``>
	RealType median(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the median of the distribution /dist/.

	[h4 [#math.dist.mode]mode]

	template<class RealType, ``__Policy``>
	RealType mode(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the mode of the distribution /dist/.

	This function may return a __domain_error if the distribution does not have
	a defined mode.

	[h4 [#math.dist.pdf]Probability Density Function]

	template <class RealType, class ``__Policy``>
	RealType pdf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);

	For a continuous function, the probability density function (pdf) returns
	the probability that the variate has the value x.
	Since for continuous distributions the probability at a single point is actually zero,
	the probability is better expressed as the integral of the pdf between two points:
	see the __cdf.

	For a discrete distribution, the pdf is the probability that the
	variate takes the value x.

	This function may return a __domain_error if the random variable is outside
	the defined range for the distribution.

	For example, for a standard normal distribution the pdf looks like this:

	[$../graphs/pdf.png]

	[h4 [#math.dist.range]Range]

	template<class RealType, class ``__Policy``>
	std::pair<RealType, RealType> range(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the valid range of the random variable over distribution /dist/.

	[h4 [#math.dist.quantile]Quantile]

	template <class RealType, class ``__Policy``>
	RealType quantile(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& p);

	The quantile is best viewed as the inverse of the __cdf, it returns
	a value /x/ such that `cdf(dist, x) == p`.

	This is also known as the /percent point function/, or a /percentile/, it is
	also the same as calculating the ['lower critical value] of a distribution.

	This function returns a __domain_error if the probability lies outside [0,1].
	The function may return an __overflow_error if there is no finite value
	that has the specified probability.

	The following graph shows the quantile function for a standard normal
	distribution:

	[$../graphs/quantile.png]

	[h4 [#math.dist.quantile_c]Quantile from the complement of the probability.]
	[link complements complements]


	template <class Distribution, class RealType>
	RealType quantile(const ``['Unspecified-Complement-Type]``<Distribution, RealType>& comp);

	This is the inverse of the __ccdf. It is calculated by wrapping
	the arguments in a call to the quantile function in a call to
	/complement/. For example:

	// define a standard normal distribution:
	boost::math::normal norm;
	// print the value of x for which the complement
	// of the probability is 0.05:
	std::cout << quantile(complement(norm, 0.05)) << std::endl;

	The function computes a value /x/ such that
	`cdf(complement(dist, x)) == q` where /q/ is complement of the
	probability.

	[link why_complements Why complements?]

	This function is also called the inverse survival function, and is the
	same as calculating the ['upper critical value] of a distribution.

	This function returns a __domain_error if the probablity lies outside [0,1].
	The function may return an __overflow_error if there is no finite value
	that has the specified probability.

	The following graph show the inverse survival function for the normal
	distribution:

	[$../graphs/survival_inv.png]

	[h4 [#math.dist.sd]Standard Deviation]

	template <class RealType, class ``__Policy``>
	RealType standard_deviation(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the standard deviation of distribution /dist/.

	This function may return a __domain_error if the distribution does not have
	a defined standard deviation.

	[h4 [#math.dist.support]support]

	template<class RealType, class ``__Policy``>
	std::pair<RealType, RealType> support(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the supported range of random variable over the distribution /dist/.

	The distribution is said to be 'supported' over a range that is
	[@http://en.wikipedia.org/wiki/Probability_distribution
	"the smallest closed set whose complement has probability zero"].
	Non-mathematicians might say it means the 'interesting' smallest range
	of random variate x that has the cdf going from zero to unity.
	Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity.

	[h4 [#math.dist.variance]Variance]

	template <class RealType, class ``__Policy``>
	RealType variance(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the variance of the distribution /dist/.

	This function may return a __domain_error if the distribution does not have
	a defined variance.

	[h4 [#math.dist.skewness]Skewness]

	template <class RealType, class ``__Policy``>
	RealType skewness(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	Returns the skewness of the distribution /dist/.

	This function may return a __domain_error if the distribution does not have
	a defined skewness.

	[h4 [#math.dist.kurtosis]Kurtosis]

	template <class RealType, class ``__Policy``>
	RealType kurtosis(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	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 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]
	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]
	of a normal distribution is zero.

	This function may return a __domain_error if the distribution does not have
	a defined kurtosis.

	'Proper' kurtosis can have a value from zero to + infinity.

	[h4 [#math.dist.kurtosis_excess]Kurtosis excess]

	template <class RealType, ``__Policy``>
	RealType kurtosis_excess(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);

	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.

	Kurtosis excess can have a value from -2 to + infinity.

	kurtosis = kurtosis_excess +3;

	The kurtosis excess of a normal distribution is zero.

	[h4 [#cdfPQ]P and Q]

	The terms P and Q are sometimes used to refer to the __cdf
	and its [link math.dist.ccdf complement] respectively.
	Lowercase p and q are sometimes used to refer to the values returned
	by these functions.

	[h4 [#percent]Percent Point Function]

	The percent point function, also known as the percentile, is the same as
	the __quantile.

	[h4 [#cdf_inv]Inverse CDF Function.]

	The inverse of the cumulative distribution function, is the same as the
	__quantile.

	[h4 [#survival_inv]Inverse Survival Function.]

	The inverse of the survival function, is the same as computing the
	[link math.dist.quantile_c quantile
	from the complement of the probability].

	[h4 [#pmf]Probability Mass Function]

	The Probability Mass Function is the same as the __pdf.

	The term Mass Function is usually applied to discrete distributions,
	while the term __pdf applies to continuous distributions.

	[h4 [#lower_critical]Lower Critical Value.]

	The lower critical value calculates the value of the random variable
	given the area under the left tail of the distribution.
	It is equivalent to calculating the __quantile.

	[h4 [#upper_critical]Upper Critical Value.]

	The upper critical value calculates the value of the random variable
	given the area under the right tail of the distribution. It is equivalent to
	calculating the [link math.dist.quantile_c quantile from the complement of the
	probability].

	[h4 [#survival]Survival Function]

	Refer to the __ccdf.

	[endsect][/section:nmp Non-Member Properties]


	[/ non_members.qbk
	Copyright 2006 John Maddock and Paul A. Bristow.
	Distributed under the Boost Software License, Version 1.0.
	(See accompanying file LICENSE_1_0.txt or copy at
	http://www.boost.org/LICENSE_1_0.txt).
	]