diff --git a/doc/sf_and_dist/Jamfile.v2 b/doc/sf_and_dist/Jamfile.v2
index 881b4e692..bd0e9d578 100644
--- a/doc/sf_and_dist/Jamfile.v2
+++ b/doc/sf_and_dist/Jamfile.v2
@@ -4,30 +4,12 @@
# file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
using quickbook ;
+using auto-index ;
+import modules ;
path-constant images_location : html ;
path-constant here : . ;
-import modules ;
-
-if --enable-index in [ modules.peek : ARGV ]
-{
- ECHO "Building the Math docs with automatic index generation enabled." ;
- using auto-index ;
- project : requirements
-
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)
@@ -104,7 +104,7 @@ are ObjectsLast revised: May 16, 2011 at 16:53:57 GMT |
+Last revised: July 16, 2011 at 10:40:56 GMT |
A B C D E F G H I K L M N P Q R S T U V W Z
acosh
+ +acoshf
+ +acoshl
+ +asinh
+ +asinhf
+ +asinhl
+ +assoc_laguerre
+ +assoc_laguerref
+ +assoc_laguerrel
+ +assoc_legendre
+ +assoc_legendref
+ +assoc_legendrel
+ +atanh
+ +atanhf
+ +atanhl
+ +beta
+ +betac
+ +betaf
+ +betal
+ +binomial_coefficient
+ +brent_find_minima
+ +Root Finding With Derivatives: Newton-Raphson, Halley & Schroeder |
cbrt
+ +cbrtf
+ +cbrtl
+ +cdf
+ +changesign
+ +checked_narrowing_cast
+ +chf
+ +comp_ellint_1
+ +comp_ellint_1f
+ +comp_ellint_1l
+ +comp_ellint_2
+ +comp_ellint_2f
+ +comp_ellint_2l
+ +comp_ellint_3
+ +comp_ellint_3f
+ +comp_ellint_3l
+ +conf_hyperg
+ +conf_hypergf
+ +conf_hypergl
+ +continued_fraction_a
+ +continued_fraction_b
+ +copysign
+ +copysignf
+ +copysignl
+ +cyl_bessel_i
+ +cyl_bessel_if
+ +cyl_bessel_il
+ +cyl_bessel_j
+ +cyl_bessel_jf
+ +cyl_bessel_jl
+ +cyl_bessel_k
+ +cyl_bessel_kf
+ +cyl_bessel_kl
+ +cyl_neumann
+ +cyl_neumannf
+ +cyl_neumannl
+ +e
+ +ellint_1
+ +ellint_1f
+ +ellint_1l
+ +ellint_2
+ +ellint_2f
+ +ellint_2l
+ +ellint_3
+ +ellint_3f
+ +ellint_3l
+ +ellint_rc
+ +ellint_rd
+ +ellint_rf
+ +ellint_rj
+ +epsilon
+ +erf
+ +erfc
+ +erfcf
+ +erfcl
+ +erfc_inv
+ +erff
+ +erfl
+ +erf_inv
+ +evaluate_even_polynomial
+ +evaluate_odd_polynomial
+ +evaluate_polynomial
+ +evaluate_rational
+ +exp2
+ +exp2f
+ +exp2l
+ +expint
+ +expintf
+ +expintl
+ +expm1
+ +expm1f
+ +expm1l
+ +e_float
+ +fdim
+ +fdimf
+ +fdiml
+ +find_beta
+ +find_degrees_of_freedom
+ +find_location
+ +find_lower_bound_on_p
+ +find_non_centrality
+ +find_scale
+ +find_upper_bound_on_p
+ +float_advance
+ +float_distance
+ +float_next
+ +float_prior
+ +fma
+ +fmaf
+ +fmal
+ +fmax
+ +fmaxf
+ +fmaxl
+ +fmin
+ +fminf
+ +fminl
+ +fpclassify
+ +gamma_p
+ +gamma_p_derivative
+ +gamma_p_inv
+ +gamma_p_inva
+ +gamma_q
+ +gamma_q_inv
+ +gamma_q_inva
+ +get_user_parameter_info
+ +halley_iterate
+ +hazard
+ +hermite
+ +hermitef
+ +hermitel
+ +hermite_next
+ +hyperg
+ +hypergf
+ +hypergl
+ +hypot
+ +hypotf
+ +hypotl
+ +Finding the Next Representable Value in a Specific Direction (nextafter) |
ibeta
+ +ibetac
+ +ibetac_inv
+ +ibetac_inva
+ +ibetac_invb
+ +ibeta_derivative
+ +ibeta_inv
+ +ibeta_inva
+ +ibeta_invb
+ +ilogb
+ +ilogbf
+ +ilogbl
+ +infinity
+ +insert
+ +iround
+ +isfinite
+ +isinf
+ +isnan
+ +isnormal
+ +itrunc
+ +kahan_sum_series
+ +kurtosis
+ +kurtosis_excess
+ +laguerre
+ +laguerref
+ +laguerrel
+ +laguerre_next
+ +ldexp
+ +legendre
+ +legendref
+ +legendrel
+ +legendre_next
+ +legendre_p
+ +legendre_q
+ +lgamma
+ +lgammaf
+ +lgammal
+ +llrint
+ +llrintf
+ +llrintl
+ +llround
+ +llroundf
+ +llroundl
+ +lltrunc
+ +log1p
+ +log1pf
+ +log1pl
+ +log2
+ +log2f
+ +log2l
+ +logb
+ +logbf
+ +logbl
+ +lrint
+ +lrintf
+ +lrintl
+ +lround
+ +lroundf
+ +lroundl
+ +ltrunc
+ +Changing the Policy on an Ad Hoc Basis for the Special Functions |
make_periodic_param
+ +make_policy
+ +make_power_param
+ +make_random_param
+ +mean
+ +median
+ +mode
+ +msg
+ +nan
+ +nanf
+ +nanl
+ +nearbyint
+ +nearbyintf
+ +nearbyintl
+ +newton_raphson_iterate
+ +nextafter
+ +nextafterf
+ +nextafterl
+ +nexttoward
+ +nexttowardf
+ +nexttowardl
+ +norm
+ +quantile
+Some Miscellaneous Examples of the Normal (Gaussian) Distribution
r
+ +range
+ +relative_error
+ +remainder
+ +remainderf
+ +remainderl
+ +remquo
+ +remquof
+ +remquol
+ +riemann_zeta
+ +riemann_zetaf
+ +riemann_zetal
+ +rint
+ +rintf
+ +rintl
+ +round
+ +roundf
+ +roundl
+ +RR
+ +scalbln
+ +scalblnf
+ +scalblnl
+ +scalbn
+ +scalbnf
+ +scalbnl
+ +scale
+schroeder_iterate
+ +shape
+ +sign
+ +signbit
+ +skewness
+ +spherical_harmonic
+ +spherical_harmonic_i
+ +spherical_harmonic_r
+ +sph_bessel
+ +sph_besself
+ +sph_bessell
+ +sph_legendre
+ +sph_legendref
+ +sph_legendrel
+ +sph_neumann
+ +sph_neumannf
+ +sph_neumannl
+ +standard_deviation
+ +sum_series
+ +t
+ +test
+ +tgamma
+Changing the Policy on an Ad Hoc Basis for the Special Functions
tgamma1pm1
+ +tgammaf
+ +tgammal
+ +tgamma_delta_ratio
+ +tgamma_lower
+ +tgamma_ratio
+ +tol
+ +trunc
+ +truncf
+ +truncl
+ +user_denorm_error
+ +user_domain_error
+ +user_evaluation_error
+ +user_indeterminate_result_error
+ +user_overflow_error
+ +user_pole_error
+ +user_rounding_error
+ +user_underflow_error
+ +value
+ +variance
+ +write_code
+ +write_csv
+ +cauchy_distribution
+ +default_policy
+ +Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
eps_tolerance
+ +equal_ceil
+ +equal_floor
+ +equal_nearest_integer
+ +fisher_f_distribution
+ +gamma_distribution
+ +inverse_gaussian_distribution
+ +log1p_series
+ +lognormal_distribution
+ +max_factorial
+ +nonfinite_num_get
+ +nonfinite_num_put
+ +non_central_beta_distribution
+ +non_central_chi_squared_distribution
+ +non_central_f_distribution
+ +non_central_t_distribution
+ +normalise
+ +normal_distribution
+ +Graphing, Profiling, and Generating Test Data for Special Functions |
test_data
+ +triangular_distribution
+ +Graphing, Profiling, and Generating Test Data for Special Functions |
upper_incomplete_gamma_fract
+ +assert_undefined_type
+ +bernoulli
+ +beta
+ +binomial
+ +denorm_error_type
+ +discrete_quantile_type
+ +domain_error_type
+ +double_t
+ +evaluation_error_type
+ +exponential
+ +extreme_value
+ +fisher_f
+ +float_t
+ +gamma
+ +geometric
+ +hypergeometric
+ +indeterminate_result_error_type
+ +inverse_chi_squared
+ +inverse_gaussian
+ +laplace
+ +logistic
+ +lognormal
+ +negative_binomial
+ +non_central_beta
+ +non_central_chi_squared
+ +non_central_f
+ +non_central_t
+ +normal
+ +overflow_error_type
+ +pareto
+ +poisson
+ +pole_error_type
+ +policy_type
+precision_type
+ +promote_double_type
+ +promote_float_type
+ +rayleigh
+ +rounding_error_type
+ +students_t
+ +triangular
+ +underflow_error_type
+ +uniform
+ +Graphing, Profiling, and Generating Test Data for Special Functions |
value_type
+Graphing, Profiling, and Generating Test Data for Special Functions
weibull
+ +BOOST_DEFINE_MATH_CONSTANT
+ +BOOST_FPU_EXCEPTION_GUARD
+ +BOOST_HAS_LOG1P
+ +BOOST_MATH_APPEND_EXPLICIT_TEMPLATE_NON_TYPE_SPEC
+ +BOOST_MATH_APPEND_EXPLICIT_TEMPLATE_TYPE_SPEC
+ +BOOST_MATH_ASSERT_UNDEFINED_POLICY
+ +BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS
+ +BOOST_MATH_CONTROL_FP
+ +BOOST_MATH_DECLARE_DISTRIBUTIONS
+ +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS
+ +BOOST_MATH_DENORM_ERROR_POLICY
+ +BOOST_MATH_DIGITS10_POLICY
+ +BOOST_MATH_DISCRETE_QUANTILE_POLICY
+ +BOOST_MATH_DOMAIN_ERROR_POLICY
+ +BOOST_MATH_EVALUATION_ERROR_POLICY
+ +BOOST_MATH_EXPLICIT_TEMPLATE_NON_TYPE
+ +BOOST_MATH_EXPLICIT_TEMPLATE_TYPE
+ +BOOST_MATH_INDETERMINATE_RESULT_ERROR_POLICY
+ +BOOST_MATH_INSTRUMENT_CODE
+ +BOOST_MATH_INSTRUMENT_FPU
+ +BOOST_MATH_INSTRUMENT_VARIABLE
+ +BOOST_MATH_INT_TABLE_TYPE
+ +BOOST_MATH_INT_VALUE_SUFFIX
+ +BOOST_MATH_MAX_POLY_ORDER
+ +BOOST_MATH_MAX_ROOT_ITERATION_POLICY
+ +BOOST_MATH_MAX_SERIES_ITERATION_POLICY
+ +BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS
+ +BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
+ +BOOST_MATH_NO_REAL_CONCEPT_TESTS
+ +BOOST_MATH_OVERFLOW_ERROR_POLICY
+ +BOOST_MATH_POLE_ERROR_POLICY
+ +BOOST_MATH_POLY_METHOD
+ +BOOST_MATH_PROMOTE_DOUBLE_POLICY
+ +BOOST_MATH_PROMOTE_FLOAT_POLICY
+ +BOOST_MATH_RATIONAL_METHOD
+ +BOOST_MATH_ROUNDING_ERROR_POLICY
+ +BOOST_MATH_SMALL_CONSTANT
+ +BOOST_MATH_STD_USING
+ +BOOST_MATH_UNDERFLOW_ERROR_POLICY
+ +BOOST_MATH_USE_C99
+ +FP_INFINITE
+ +FP_NAN
+ +FP_NORMAL
+ +FP_SUBNORMAL
+ +FP_ZERO
+ +acosh
+ +acoshf
+ +acoshl
+ +Advancing a Floating Point Value by a Specific Representation Distance (ULP) float_advance
+ +asinh
+ +asinhf
+ +asinhl
+ +assert_undefined_type
+ +assoc_laguerre
+ +assoc_laguerref
+ +assoc_laguerrel
+ +assoc_legendre
+ +assoc_legendref
+ +assoc_legendrel
+ +atanh
+ +atanhf
+ +atanhl
+ +bernoulli
+ +Bernoulli Distribution
+ +Bessel Functions of the First and Second Kinds
+ +Beta
+ +beta
+ +Beta Distribution
+ +betac
+ +betaf
+ +betal
+ +binomial
+ +Binomial Coefficients
+ +Binomial Distribution
+ +binomial_coefficient
+ +Boost.Math Macros
+Boost.Math Tuning
+ +BOOST_DEFINE_MATH_CONSTANT
+ +BOOST_FPU_EXCEPTION_GUARD
+ +BOOST_HAS_LOG1P
+ +BOOST_MATH_APPEND_EXPLICIT_TEMPLATE_NON_TYPE_SPEC
+ +BOOST_MATH_APPEND_EXPLICIT_TEMPLATE_TYPE_SPEC
+ +BOOST_MATH_ASSERT_UNDEFINED_POLICY
+ +BOOST_MATH_BUGGY_LARGE_FLOAT_CONSTANTS
+ +BOOST_MATH_CONTROL_FP
+ +BOOST_MATH_DECLARE_DISTRIBUTIONS
+ +BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS
+ +BOOST_MATH_DENORM_ERROR_POLICY
+ +BOOST_MATH_DIGITS10_POLICY
+ +BOOST_MATH_DISCRETE_QUANTILE_POLICY
+ +BOOST_MATH_DOMAIN_ERROR_POLICY
+ +BOOST_MATH_EVALUATION_ERROR_POLICY
+ +BOOST_MATH_EXPLICIT_TEMPLATE_NON_TYPE
+ +BOOST_MATH_EXPLICIT_TEMPLATE_TYPE
+ +BOOST_MATH_INDETERMINATE_RESULT_ERROR_POLICY
+ +BOOST_MATH_INSTRUMENT_CODE
+ +BOOST_MATH_INSTRUMENT_FPU
+ +BOOST_MATH_INSTRUMENT_VARIABLE
+ +BOOST_MATH_INT_TABLE_TYPE
+ +BOOST_MATH_INT_VALUE_SUFFIX
+ +BOOST_MATH_MAX_POLY_ORDER
+ +BOOST_MATH_MAX_ROOT_ITERATION_POLICY
+ +BOOST_MATH_MAX_SERIES_ITERATION_POLICY
+ +BOOST_MATH_NO_DEDUCED_FUNCTION_POINTERS
+ +BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
+ +BOOST_MATH_NO_REAL_CONCEPT_TESTS
+ +BOOST_MATH_OVERFLOW_ERROR_POLICY
+ +BOOST_MATH_POLE_ERROR_POLICY
+ +BOOST_MATH_POLY_METHOD
+ +BOOST_MATH_PROMOTE_DOUBLE_POLICY
+ +BOOST_MATH_PROMOTE_FLOAT_POLICY
+ +BOOST_MATH_RATIONAL_METHOD
+ +BOOST_MATH_ROUNDING_ERROR_POLICY
+ +BOOST_MATH_SMALL_CONSTANT
+ +BOOST_MATH_STD_USING
+ +BOOST_MATH_UNDERFLOW_ERROR_POLICY
+ +BOOST_MATH_USE_C99
+ +brent_find_minima
+ +Root Finding With Derivatives: Newton-Raphson, Halley & Schroeder |
C99 and C++ TR1 C-style Functions
+C99 and TR1 C Functions Overview
+C99 C Functions
+Calling User Defined Error Handlers
+ +cauchy
+ +Cauchy-Lorentz Distribution
+ +cauchy_distribution
+ +cbrt
+ +cbrtf
+ +cbrtl
+ +cdf
+ +changesign
+ +Changing the Policy Defaults
+ +Changing the Policy on an Ad Hoc Basis for the Special Functions
+ +checked_narrowing_cast
+ +chf
+ +Chi Squared Distribution
+ +chi_squared
+ +Compile Time Power of a Runtime Base
+ +Compilers
+ +Complements are supported too - and when to use them
+ +comp_ellint_1
+ +comp_ellint_1f
+ +comp_ellint_1l
+ +comp_ellint_2
+ +comp_ellint_2f
+ +comp_ellint_2l
+ +comp_ellint_3
+ +comp_ellint_3f
+ +comp_ellint_3l
+ +Conceptual Requirements for Distribution Types
+ +Conceptual Requirements for Real Number Types
+ +confidence intervals on the mean with the Students-t distribution
+ +conf_hyperg
+ +conf_hypergf
+ +conf_hypergl
+ +Continued Fraction Evaluation
+ +continued_fraction_a
+ +continued_fraction_b
+ +copysign
+ +copysignf
+ +copysignl
+ +cyl_bessel_i
+ +cyl_bessel_if
+ +cyl_bessel_il
+ +cyl_bessel_j
+ +cyl_bessel_jf
+ +cyl_bessel_jl
+ +cyl_bessel_k
+ +cyl_bessel_kf
+ +cyl_bessel_kl
+ +cyl_neumann
+ +cyl_neumannf
+ +cyl_neumannl
+ +default_policy
+ +denorm_error_type
+ +Derivative of the Incomplete Beta Function
+ +Derivative of the Incomplete Gamma Function
+ +Digamma
+ +Discrete Quantile Policies
+ +discrete_quantile_type
+ +Distribution Algorithms
+ +domain_error_type
+ +double_t
+ +Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
Root Finding Without Derivatives: Bisection, Bracket and TOMS748 |
e
+ +ellint_1
+ +ellint_1f
+ +ellint_1l
+ +ellint_2
+ +ellint_2f
+ +ellint_2l
+ +ellint_3
+ +ellint_3f
+ +ellint_3l
+ +ellint_rc
+ +ellint_rd
+ +ellint_rf
+ +ellint_rj
+ +Elliptic Integrals - Carlson Form
+ +Elliptic Integrals of the First Kind - Legendre Form
+ +Elliptic Integrals of the Second Kind - Legendre Form
+ +Elliptic Integrals of the Third Kind - Legendre Form
+ +epsilon
+ +eps_tolerance
+ +equal_ceil
+ +equal_floor
+ +equal_nearest_integer
+ +erf
+ +erfc
+ +erfcf
+ +erfcl
+ +erfc_inv
+ +erff
+ +erfl
+ +erf_inv
+ +Error Function Inverses
+ +Error Functions
+ +Error Handling
+ +Error Handling Policies
+ +Errors In the Function beta(a, b, x)
+ +Errors In the Function betac(a,b,x)
+ +Errors In the Function erf(z)
+ +Errors In the Function erfc(z)
+ +Errors In the Function expint(n, z)
+ +Errors In the Function expint(z)
+ +Errors In the Function gamma_p(a,z)
+ +Errors In the Function gamma_q(a,z)
+ +Errors In the Function ibeta(a,b,x)
+ +Errors In the Function ibetac(a,b,x)
+ +Errors In the Function tgamma(a,z)
+ +Errors In the Function tgamma_delta_ratio(a, delta)
+ +Errors In the Function tgamma_lower(a,z)
+ +Errors In the Function tgamma_ratio(a, b)
+ +Errors In the Function zeta(z)
+ +evaluate_even_polynomial
+ +evaluate_odd_polynomial
+ +evaluate_polynomial
+ +evaluate_rational
+ +evaluation_error_type
+ +Examples
+ +exp2
+ +exp2f
+ +exp2l
+ +expint
+ +expintf
+ +expintl
+ +expm1
+ +expm1f
+ +expm1l
+ +exponential
+ +Exponential Distribution
+ +Exponential Integral Ei
+ +Exponential Integral En
+ +Extras/Future Directions
+ +Extreme Value Distribution
+ +extreme_value
+ +e_float
+ +e_float Support
+ +F Distribution
+ +Facets for Floating-Point Infinities and NaNs
+ +Factorial
+ +fdim
+ +fdimf
+ +fdiml
+ +Finding the Next Greater Representable Value (float_next)
+ +Finding the Next Representable Value in a Specific Direction (nextafter)
+ +Finding the Next Smaller Representable Value (float_prior)
+ +find_beta
+ +find_degrees_of_freedom
+ +find_location
+ +find_lower_bound_on_p
+ +find_non_centrality
+ +find_scale
+ +find_upper_bound_on_p
+ +fisher_f
+ +fisher_f_distribution
+ +Floating-Point Classification: Infinities and NaNs
+float_advance
+ +float_distance
+ +float_next
+ +float_prior
+ +float_t
+ +fma
+ +fmaf
+ +fmal
+ +fmax
+ +fmaxf
+ +fmaxl
+ +fmin
+ +fminf
+ +fminl
+ +fpclassify
+ +FP_INFINITE
+ +FP_NAN
+ +FP_NORMAL
+ +FP_SUBNORMAL
+ +FP_ZERO
+ +Gamma
+ +gamma
+ +Gamma (and Erlang) Distribution
+ +gamma_distribution
+ +gamma_p
+ +gamma_p_derivative
+ +gamma_p_inv
+ +gamma_p_inva
+ +gamma_q
+ +gamma_q_inv
+ +gamma_q_inva
+ +Generic operations common to all distributions are non-member functions
+ +geometric
+ +Geometric Distribution
+ +get_user_parameter_info
+ +Graphing, Profiling, and Generating Test Data for Special Functions
+ +halley_iterate
+ +hazard
+ +hermite
+ +Hermite Polynomials
+hermitef
+ +hermitel
+ +hermite_next
+ +History and What's New
+ +hyperg
+ +hypergeometric
+ +Hypergeometric Distribution
+ +hypergf
+ +hypergl
+ +hypot
+ +hypotf
+ +hypotl
+ +ibeta
+ +ibetac
+ +ibetac_inv
+ +ibetac_inva
+ +ibetac_invb
+ +ibeta_derivative
+ +ibeta_inv
+ +ibeta_inva
+ +ibeta_invb
+ +ilogb
+ +ilogbf
+ +ilogbl
+ +Implementation Notes
+ +Incomplete Beta Function Inverses
+ +Incomplete Beta Functions
+ +Incomplete Gamma Function Inverses
+ +Incomplete Gamma Functions
+ +indeterminate_result_error_type
+ +infinity
+ +insert
+ +Introduction
+ +Inverse Chi Squared Distribution
+ +Inverse Gamma Distribution
+ +Inverse Gaussian (or Inverse Normal) Distribution
+ +inverse_chi_squared
+ +inverse_gaussian
+ +inverse_gaussian_distribution
+ +iround
+ +isfinite
+ +isinf
+ +isnan
+ +isnormal
+ +Iteration Limits Policies
+ +itrunc
+ +kahan_sum_series
+ +Known Issues, and TODO List
+ +kurtosis
+ +kurtosis_excess
+ +Using With MPFR / GMP - a High-Precision Floating-Point Library |
laguerre
+ +Laguerre (and Associated) Polynomials
+ +laguerref
+ +laguerrel
+ +laguerre_next
+ +Lanczos approximation
+Using With MPFR / GMP - a High-Precision Floating-Point Library
laplace
+ +Laplace Distribution
+ +ldexp
+ +legendre
+ +Legendre (and Associated) Polynomials
+ +legendref
+ +legendrel
+ +legendre_next
+ +legendre_p
+ +legendre_q
+ +lgamma
+ +lgammaf
+ +lgammal
+ +llrint
+ +llrintf
+ +llrintl
+ +llround
+ +llroundf
+ +llroundl
+ +lltrunc
+ +Locating Function Minima: Brent's algorithm
+ +Log Gamma
+ +Log Normal Distribution
+ +log1p
+ +log1pf
+ +log1pl
+ +log1p_series
+ +log2
+ +log2f
+ +log2l
+ +logb
+ +logbf
+ +logbl
+ +logistic
+ +Logistic Distribution
+ +lognormal
+ +lognormal_distribution
+ +lrint
+ +lrintf
+ +lrintl
+ +lround
+ +lroundf
+ +lroundl
+ +ltrunc
+ +make_periodic_param
+ +make_policy
+ +make_power_param
+ +make_random_param
+ +Mathematically Undefined Function Policies
+ +max_factorial
+ +mean
+ +median
+ +mode
+ +Modified Bessel Functions of the First and Second Kinds
+ +msg
+ +Finding the Next Representable Value in a Specific Direction (nextafter) |
Namespaces
+ +nan
+ +nanf
+ +nanl
+ +nearbyint
+ +nearbyintf
+ +nearbyintl
+ +Negative Binomial Distribution
+ +negative_binomial
+ +newton_raphson_iterate
+ +nextafter
+ +nextafterf
+ +nextafterl
+ +nexttoward
+ +nexttowardf
+ +nexttowardl
+ +Non-Member Properties
+ +Noncentral Beta Distribution
+ +Noncentral Chi-Squared Distribution
+ +Noncentral F Distribution
+ +Noncentral T Distribution
+ +nonfinite_num_get
+ +nonfinite_num_put
+ +non_central_beta
+ +non_central_beta_distribution
+ +non_central_chi_squared
+ +non_central_chi_squared_distribution
+ +non_central_f
+ +non_central_f_distribution
+ +non_central_t
+ +non_central_t_distribution
+ +norm
+ +normal
+ +Normal (Gaussian) Distribution
+ +normalise
+ +normal_distribution
+ +Numeric Constants
+ +overflow_error_type
+ +pareto
+ +Pareto Distribution
+ +Performance Tuning Macros
+ +poisson
+ +Poisson Distribution
+ +pole_error_type
+ +Policy Class Reference
+policy_type
+Polynomial and Rational Function Evaluation
+ +Polynomials
+ +precision_type
+ +promote_args
+ +promote_double_type
+ +promote_float_type
+ +quantile
+Some Miscellaneous Examples of the Normal (Gaussian) Distribution
r
+ +range
+ +Ratios of Gamma Functions
+ +rayleigh
+ +Rayleigh Distribution
+ +Reference
+ +References
+ +Relative Error and Testing
+relative_error
+ +remainder
+ +remainderf
+ +remainderl
+ +remquo
+ +remquof
+ +remquol
+ +Representation Distance Between Two Floating Point Values (ULP) float_distance
+Riemann Zeta Function
+ +riemann_zeta
+ +riemann_zetaf
+ +riemann_zetal
+ +rint
+ +rintf
+ +rintl
+ +Root Finding With Derivatives: Newton-Raphson, Halley & Schroeder
+ +Root Finding Without Derivatives: Bisection, Bracket and TOMS748
+ +round
+ +roundf
+ +Rounding Functions
+ +rounding_error_type
+ +roundl
+ +RR
+ +Graphing, Profiling, and Generating Test Data for Special Functions |
Changing the Policy on an Ad Hoc Basis for the Special Functions |
scalbln
+ +scalblnf
+ +scalblnl
+ +scalbn
+ +scalbnf
+ +scalbnl
+ +scale
+schroeder_iterate
+ +Series Evaluation
+ +Setting Polices at Namespace Scope
+ +Setting Policies at Namespace or Translation Unit Scope
+ +Setting Policies for Distributions on an Ad Hoc Basis
+ +shape
+ +sign
+ +Sign Manipulation Functions
+signbit
+ +skewness
+ +Spherical Bessel Functions of the First and Second Kinds
+ +Spherical Harmonics
+ +spherical_harmonic
+ +spherical_harmonic_i
+ +spherical_harmonic_r
+ +sph_bessel
+ +sph_besself
+ +sph_bessell
+ +sph_legendre
+ +sph_legendref
+ +sph_legendrel
+ +sph_neumann
+ +sph_neumannf
+ +sph_neumannl
+ +standard_deviation
+ +Students t Distribution
+ +students_t
+ +sum_series
+ +Supported/Tested Compilers
+ +Graphing, Profiling, and Generating Test Data for Special Functions |
t
+ +test
+ +test_data
+ +tgamma
+Changing the Policy on an Ad Hoc Basis for the Special Functions
tgamma1pm1
+ +tgammaf
+ +tgammal
+ +tgamma_delta_ratio
+ +tgamma_lower
+ +tgamma_ratio
+ +tol
+ +TR1 C Functions Quick Reference
+triangular
+ +Triangular Distribution
+ +triangular_distribution
+ +trunc
+ +Truncation Functions
+ +truncf
+ +truncl
+ +Graphing, Profiling, and Generating Test Data for Special Functions |
underflow_error_type
+ +uniform
+ +Uniform Distribution
+ +upper_incomplete_gamma_fract
+ +user_denorm_error
+ +user_domain_error
+ +user_evaluation_error
+ +user_indeterminate_result_error
+ +user_overflow_error
+ +user_pole_error
+ +user_rounding_error
+ +user_underflow_error
+ +Using Macros to Change the Policy Defaults
+Using With MPFR / GMP - a High-Precision Floating-Point Library
+ +Using With NTL - a High-Precision Floating-Point Library
+ +value
+ +value_type
+Graphing, Profiling, and Generating Test Data for Special Functions
variance
+ +weibull
+ +Weibull Distribution
+ +write_code
+ +write_csv
+ +@@ -99,7 +99,7 @@ functions divided by large powers into single (simpler) expressions.
@@ -120,7 +120,7 @@ Binomial Probability Distribution Calculator.
diff --git a/doc/sf_and_dist/html/math_toolkit/backgrounders/relative_error.html b/doc/sf_and_dist/html/math_toolkit/backgrounders/relative_error.html index e2e2caaf7..5b0f742a5 100644 --- a/doc/sf_and_dist/html/math_toolkit/backgrounders/relative_error.html +++ b/doc/sf_and_dist/html/math_toolkit/backgrounders/relative_error.html @@ -78,7 +78,7 @@ of binary digits. You have been warned!
@@ -250,7 +250,7 @@
diff --git a/doc/sf_and_dist/html/math_toolkit/dist.html b/doc/sf_and_dist/html/math_toolkit/dist.html index a97c7ccd1..ab5eb8ba6 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist.html @@ -38,7 +38,7 @@ are Objects
@@ -136,7 +136,7 @@ and so should have errors within an epsilon or two.
@@ -327,7 +327,7 @@
beta_distribution(RealType alpha, RealType beta); @@ -164,7 +164,7 @@ yellow in the graph above).- + Parameter Accessors
@@ -188,7 +188,7 @@ assert(mybeta.beta() == 5.); // mybeta.beta() returns 5
cdf(beta_distribution<RealType>(alpha, beta), x) == probability.
@@ -276,7 +276,7 @@ statistical inference.
@@ -315,7 +315,7 @@ please refer to these functions for information on accuracy.
@@ -594,7 +594,7 @@
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html index 1973c364c..126139994 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/binomial_dist.html @@ -161,12 +161,12 @@
binomial_distribution(RealType n, RealType p); @@ -183,7 +183,7 @@ otherwise calls domain_error.- + Accessors
RealType success_fraction() const; @@ -199,7 +199,7 @@ was constructed.- + Lower Bound on the Success Fraction
@@ -305,7 +305,7 @@ limits illustrated in the case of the binomial. Biometrika 26 404-413.- + Upper Bound on the Success Fraction
@@ -383,7 +383,7 @@- + Estimating the Number of Trials Required for a Certain Number of Successes
@@ -425,7 +425,7 @@ of seeing 10 events that occur with frequency one half.- + Estimating the Maximum Number of Trials to Ensure no more than a Certain Number of Successes @@ -473,7 +473,7 @@ Worked Example.
- + Non-member Accessors
@@ -622,7 +622,7 @@- + Examples
@@ -630,7 +630,7 @@ examples are available illustrating the use of the binomial distribution.
- + Accuracy
@@ -640,7 +640,7 @@ please refer to these functions for information on accuracy.
- + Implementation
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html index 010654627..e4990a1f0 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/cauchy_dist.html @@ -88,7 +88,7 @@
![]()
- + Member Functions
@@ -114,7 +114,7 @@ Returns the scale parameter of the distribution.- + Non-member Accessors
@@ -148,7 +148,7 @@ The domain of the random variable is [-[max_value], +[min_value]].- + Accuracy
@@ -157,7 +157,7 @@ have very low error rates.
- + Implementation
@@ -273,7 +273,7 @@
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html index 25e1a6c27..c8a221fe1 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/chi_squared_dist.html @@ -87,7 +87,7 @@ independent, normally distributed random
![]()
- + Member Functions
@@ -170,7 +170,7 @@ independent, normally distributed random NIST Engineering Statistics Handbook, Section 7.2.3.2.- + Non-member Accessors
@@ -196,7 +196,7 @@ independent, normally distributed random The domain of the random variable is [0, +∞].- + Examples
@@ -204,7 +204,7 @@ independent, normally distributed random are available illustrating the use of the Chi Squared Distribution.
- + Accuracy
@@ -212,7 +212,7 @@ independent, normally distributed random gamma functions: please refer to the accuracy data for those functions.
- + Implementation
@@ -379,7 +379,7 @@ independent, normally distributed random
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/exp_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/exp_dist.html index 085abbf90..4b4f17712 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/exp_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/exp_dist.html @@ -71,7 +71,7 @@
![]()
- + Member Functions
@@ -91,7 +91,7 @@ Accessor function returns the lambda parameter of the distribution.- + Non-member Accessors
@@ -111,7 +111,7 @@ The domain of the random variable is [0, +∞].- + Accuracy
@@ -122,7 +122,7 @@ should have very low error rates.
- + Implementation
@@ -283,7 +283,7 @@
- + references
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html index 4612ad54c..8d3186d10 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/extreme_dist.html @@ -100,7 +100,7 @@
![]()
- + Member Functions
@@ -125,7 +125,7 @@ Returns the scale parameter of the distribution.- + Non-member Accessors
@@ -145,7 +145,7 @@ The domain of the random parameter is [-∞, +∞].- + Accuracy
@@ -154,7 +154,7 @@ very low error rates.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/f_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/f_dist.html index 17de4917e..e730c6d17 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/f_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/f_dist.html @@ -80,7 +80,7 @@
![]()
- + Member Functions
@@ -106,7 +106,7 @@ Returns the denominator degrees of freedom parameter of the distribution.- + Non-member Accessors
@@ -126,7 +126,7 @@ The domain of the random variable is [0, +∞].- + Examples
@@ -134,7 +134,7 @@ are available illustrating the use of the F Distribution.
- + Accuracy
@@ -143,7 +143,7 @@ refer to those functions for accuracy data.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html index 559e95cbc..de1ae5980 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html @@ -137,7 +137,7 @@ than a dedicated Erlang Distribution.
- + Member Functions
@@ -162,7 +162,7 @@ Returns the scale parameter of this distribution.- + Non-member Accessors
@@ -182,7 +182,7 @@ The domain of the random variable is [0,+∞].- + Accuracy
@@ -194,7 +194,7 @@ refer to the accuracy data for those functions for more information.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html index 89fff8778..1090a4337 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/geometric_dist.html @@ -146,7 +146,7 @@
![]()
- + Related Distributions
@@ -206,12 +206,12 @@- + Member Functions
- + Constructor
geometric_distribution(RealType p); @@ -226,7 +226,7 @@ 1.- + Accessors
RealType success_fraction() const; // successes / trials (0 <= p <= 1) @@ -253,7 +253,7 @@ Binomial Distribution for more discussion.- + Lower Bound on success_fraction Parameter p
@@ -308,7 +308,7 @@ vol. 48, no3, 605-621.- + Upper Bound on success_fraction Parameter p
@@ -363,7 +363,7 @@ vol. 48, no3, 605-621.- + Estimating Number of Trials to Ensure at Least a Certain Number of Failures
@@ -415,7 +415,7 @@ probability of observing k failures or fewer.- + Estimating Number of Trials to Ensure a Maximum Number of Failures or Less
@@ -463,7 +463,7 @@ probability of observing more than k failures.- + Non-member Accessors
@@ -611,7 +611,7 @@- + Accuracy
@@ -622,7 +622,7 @@ for example to 10 decimal digits (from 16).
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/hypergeometric_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/hypergeometric_dist.html index 0a586f7ab..c61abc2e4 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/hypergeometric_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/hypergeometric_dist.html @@ -102,7 +102,7 @@
![]()
- + Member Functions
@@ -131,7 +131,7 @@ from the population N.- + Non-member Accessors
@@ -185,7 +185,7 @@- + Accuracy
@@ -211,7 +211,7 @@ meaningless for N >= 1015.
- + Testing
@@ -223,7 +223,7 @@ this implementation and NTL::RR.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html index dd15e5f6a..c1d81c88b 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html @@ -198,7 +198,7 @@
![]()
- + Member Functions
@@ -225,7 +225,7 @@ Returns the scale ξ parameter of this distribution.- + Non-member Accessors
@@ -255,7 +255,7 @@- + Accuracy
@@ -271,7 +271,7 @@ iteration is involved, as for the estimation of degrees of freedom.
- + Implementation
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gamma_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gamma_dist.html index abaa38cf1..3ba29e37f 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gamma_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gamma_dist.html @@ -130,7 +130,7 @@
![]()
- + Member Functions
@@ -154,7 +154,7 @@ Returns the β scale parameter of this inverse gamma distribution.- + Non-member Accessors
@@ -184,7 +184,7 @@- + Accuracy
@@ -198,7 +198,7 @@ >14 decimal digits accuracy for 64-bit double.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html index e8fcb2c8c..414504f3d 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html @@ -146,7 +146,7 @@ the __wald_distrib (where mean μ is unity) is also provided.
- + Member Functions
@@ -171,7 +171,7 @@ Returns the scale λ parameter of this distribution.- + Non-member Accessors
@@ -201,7 +201,7 @@- + Accuracy
@@ -212,7 +212,7 @@ to a few epsilon, >14 decimal digits accuracy for 64-bit double.
- + Implementation
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/laplace_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/laplace_dist.html index c064fba5e..eb21847d8 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/laplace_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/laplace_dist.html @@ -81,7 +81,7 @@
![]()
- + Member Functions
@@ -113,7 +113,7 @@ Returns the scale parameter of this distribution.- + Non-member Accessors
@@ -133,7 +133,7 @@ The domain of the random variable is [-∞,+∞].- + Accuracy
@@ -141,7 +141,7 @@ log and exp functions and as such should have very small errors.
- + Implementation
@@ -329,7 +329,7 @@
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/logistic_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/logistic_dist.html index e5f34f7b0..b204d2aba 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/logistic_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/logistic_dist.html @@ -72,7 +72,7 @@
![]()
- + Member Functions
@@ -98,7 +98,7 @@ Returns the scale of this distribution.- + Non-member Accessors
@@ -128,7 +128,7 @@ as special cases if RealType and +overflow_error respectively.- + Accuracy
@@ -140,7 +140,7 @@ as special cases if RealType error can be guarenteed.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html index c29a89d40..adc910c56 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/lognormal_dist.html @@ -88,7 +88,7 @@
![]()
- + Member Functions
@@ -121,7 +121,7 @@ Returns the scale parameter of this distribution.- + Non-member Accessors
@@ -141,7 +141,7 @@ The domain of the random variable is [0,+∞].- + Accuracy
@@ -150,7 +150,7 @@ function, and as such should have very low error rates.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_beta_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_beta_dist.html index 7fd4c5e6c..92a083164 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_beta_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_beta_dist.html @@ -96,7 +96,7 @@ is a central χ2 random variable with
![]()
- + Member Functions
@@ -128,7 +128,7 @@ is a central χ2 random variable with was constructed.- + Non-member Accessors
@@ -156,7 +156,7 @@ is a central χ2 random variable with The domain of the random variable is [0, 1].- + Accuracy
@@ -299,7 +299,7 @@ is a central χ2 random variable with functions are broadly similar.
- + Tests
@@ -311,7 +311,7 @@ is a central χ2 random variable with tests.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_chi_squared_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_chi_squared_dist.html index dbc6a232a..315b40af1 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_chi_squared_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_chi_squared_dist.html @@ -110,7 +110,7 @@
![]()
- + Member Functions
@@ -183,7 +183,7 @@ == q.- + Non-member Accessors
@@ -203,7 +203,7 @@ The domain of the random variable is [0, +∞].- + Examples
@@ -211,7 +211,7 @@ example for the noncentral chi-squared distribution.
- + Accuracy
@@ -359,7 +359,7 @@ produce an accuracy greater than the square root of the machine epsilon.
- + Tests
@@ -373,7 +373,7 @@ to at least 50 decimal digits - and is the used for our accuracy tests.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html index 681c6faa0..f1130daee 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_f_dist.html @@ -95,7 +95,7 @@ random variable with v1
![]()
- + Member Functions
@@ -127,7 +127,7 @@ random variable with v1 which this object was constructed.- + Non-member Accessors
@@ -147,7 +147,7 @@ random variable with v1 The domain of the random variable is [0, +∞].- + Accuracy
@@ -155,7 +155,7 @@ random variable with v1 Beta Distribution: refer to that distribution for accuracy data.
- + Tests
@@ -164,7 +164,7 @@ random variable with v1 Math library statistical package and its pbeta and dbeta functions.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_t_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_t_dist.html index 11a801743..28b890256 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_t_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/nc_t_dist.html @@ -85,7 +85,7 @@
![]()
- + Member Functions
@@ -111,7 +111,7 @@ which this object was constructed.- + Non-member Accessors
@@ -131,7 +131,7 @@ The domain of the random variable is [-∞, +∞].- + Accuracy
@@ -255,7 +255,7 @@ epsilon.
- + Tests
@@ -270,7 +270,7 @@ least 50 decimal places.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html index 446ac4291..05b91eaed 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/negative_binomial_dist.html @@ -123,7 +123,7 @@
![]()
- + Related Distributions
@@ -195,12 +195,12 @@- + Member Functions
- + Construct
negative_binomial_distribution(RealType r, RealType p); @@ -216,7 +216,7 @@ <= 1.- + Accessors
RealType success_fraction() const; // successes / trials (0 <= p <= 1) @@ -237,7 +237,7 @@ Distribution for more discussion.- + Lower Bound on Parameter p
@@ -298,7 +298,7 @@ vol. 48, no3, 605-621.- + Upper Bound on Parameter p
@@ -358,7 +358,7 @@ vol. 48, no3, 605-621.- + Estimating Number of Trials to Ensure at Least a Certain Number of Failures
@@ -409,7 +409,7 @@ probability of observing k failures or fewer.- + Estimating Number of Trials to Ensure a Maximum Number of Failures or Less
@@ -457,7 +457,7 @@ probability of observing more than k failures.- + Non-member Accessors
@@ -606,7 +606,7 @@- + Accuracy
@@ -616,7 +616,7 @@ please refer to these functions for information on accuracy.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/normal_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/normal_dist.html index 9dd7fbd9d..c48804071 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/normal_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/normal_dist.html @@ -79,7 +79,7 @@
![]()
- + Member Functions
@@ -109,7 +109,7 @@ be used generically).- + Non-member Accessors
@@ -131,7 +131,7 @@ and complement cdf -∞ = 1 and +∞ = 0, if RealType permits.- + Accuracy
@@ -139,7 +139,7 @@ function, and as such should have very low error rates.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/pareto.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/pareto.html index b45d1e139..7389542a6 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/pareto.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/pareto.html @@ -86,12 +86,12 @@
![]()
- + Related distributions
- + Member Functions
@@ -117,7 +117,7 @@ Returns the shape parameter of this distribution.- + Non-member Accessors
@@ -137,7 +137,7 @@ The supported domain of the random variable is [scale, ∞].- + Accuracy
@@ -150,7 +150,7 @@ zero) see also why complements?.
- + Implementation
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html index 85c6bfeef..6be378085 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/poisson_dist.html @@ -116,7 +116,7 @@
- + Member Functions
@@ -131,7 +131,7 @@ Returns the mean of this distribution.- + Non-member Accessors
@@ -151,7 +151,7 @@ The domain of the random variable is [0, ∞].- + Accuracy
@@ -165,7 +165,7 @@ using an iterative method with a lower tolerance to avoid excessive computation.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/rayleigh.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/rayleigh.html index 7a2633c7f..652812e7b 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/rayleigh.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/rayleigh.html @@ -86,7 +86,7 @@
![]()
- + Related distributions
@@ -102,7 +102,7 @@ distribution.- + Member Functions
@@ -121,7 +121,7 @@ Returns the sigma parameter of this distribution.- + Non-member Accessors
@@ -141,7 +141,7 @@ The domain of the random variable is [0, max_value].- + Accuracy
@@ -151,7 +151,7 @@ using NTL RR type with 150-bit accuracy, about 50 decimal digits.
- + Implementation
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html index f8ed15f76..c3c34d642 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/students_t_dist.html @@ -97,7 +97,7 @@
![]()
- + Member Functions
@@ -174,7 +174,7 @@ Engineering Statistics Handbook.- + Non-member Accessors
@@ -194,7 +194,7 @@ The domain of the random variable is [-∞, +∞].- + Examples
@@ -202,7 +202,7 @@ are available illustrating the use of the Student's t distribution.
- + Accuracy
@@ -211,7 +211,7 @@ inverses, refer to accuracy data on those functions for more information.
- + Implementation
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html index 3a5386d1f..59905649b 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/triangular_dist.html @@ -128,7 +128,7 @@
![]()
- + Member Functions
@@ -163,7 +163,7 @@ (default+1).- + Non-member Accessors
@@ -184,7 +184,7 @@ range is lower <= x <= upper.- + Accuracy
@@ -193,7 +193,7 @@ with arguments nearing the extremes of zero and unity.
- + Implementation
@@ -378,7 +378,7 @@ Calculate and plot probability distributions
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html index cb56cd703..ae855070b 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/uniform_dist.html @@ -117,7 +117,7 @@
![]()
- + Member Functions
@@ -144,7 +144,7 @@ Returns the upper parameter of this distribution.- + Non-member Accessors
@@ -165,7 +165,7 @@ range is only lower <= x <= upper.- + Accuracy
@@ -173,7 +173,7 @@ and so should have errors within an epsilon or two.
- + Implementation
@@ -337,7 +337,7 @@
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html index 27813ce4a..229d312a4 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/weibull_dist.html @@ -100,7 +100,7 @@
![]()
- + Related distributions
@@ -114,7 +114,7 @@ Distributions, Theory and Applications Samuel Kotz & Saralees Nadarajah.- + Member Functions
@@ -140,7 +140,7 @@ Returns the scale parameter of this distribution.- + Non-member Accessors
@@ -160,7 +160,7 @@ The domain of the random variable is [0, ∞].- + Accuracy
@@ -170,7 +170,7 @@ as such should have very low error rates.
- + Implementation
@@ -337,7 +337,7 @@
- + References
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/nmp.html b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/nmp.html index c36ba88f6..6a0e54255 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/nmp.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/nmp.html @@ -37,7 +37,7 @@ the function you want if you already know its name.
- + Function Index
@@ -94,7 +94,7 @@
- + Conceptual Index
@@ -180,7 +180,7 @@
- + Cumulative Distribution Function
template <class RealType, class Policy> @@ -203,7 +203,7 @@![]()
- + Complement of the Cumulative Distribution Function
template <class Distribution, class RealType> @@ -243,7 +243,7 @@ complement is useful and when it should be used.- + Hazard Function
template <class RealType, class Policy> @@ -271,7 +271,7 @@- + Cumulative Hazard Function
template <class RealType, class Policy> @@ -298,7 +298,7 @@- + mean
template<class RealType, class Policy> @@ -313,7 +313,7 @@ distribution).- + median
template<class RealType, class Policy> @@ -323,7 +323,7 @@ Returns the median of the distribution dist.- + mode
template<class RealType, Policy> @@ -337,7 +337,7 @@ if the distribution does not have a defined mode.- + Probability Density Function
template <class RealType, class Policy> @@ -365,7 +365,7 @@![]()
- + Range
template<class RealType, class Policy> @@ -375,7 +375,7 @@ Returns the valid range of the random variable over distribution dist.- + Quantile
template <class RealType, class Policy> @@ -405,7 +405,7 @@![]()
- + Quantile from the complement of the probability.
@@ -450,7 +450,7 @@
![]()
- + Standard Deviation
template <class RealType, class Policy> @@ -464,7 +464,7 @@ if the distribution does not have a defined standard deviation.- + support
template<class RealType, class Policy> @@ -481,7 +481,7 @@ where the pdf is zero, and the cdf zero or unity.- + Variance
template <class RealType, class Policy> @@ -495,7 +495,7 @@ if the distribution does not have a defined variance.- + Skewness
template <class RealType, class Policy> @@ -509,7 +509,7 @@ if the distribution does not have a defined skewness.- + Kurtosis
template <class RealType, class Policy> @@ -551,7 +551,7 @@ 'Proper' kurtosis can have a value from zero to + infinity.- + Kurtosis excess
template <class RealType, Policy> @@ -585,7 +585,7 @@ The kurtosis excess of a normal distribution is zero.- + P and Q
@@ -595,7 +595,7 @@ returned by these functions.
- + Percent Point Function or Percentile
@@ -603,7 +603,7 @@ the Quantile.
- + Inverse CDF Function.
@@ -611,14 +611,14 @@ Quantile.
- + Inverse Survival Function.
The inverse of the survival function, is the same as computing the quantile from the complement of the probability.
- + Probability Mass Function
@@ -631,7 +631,7 @@ applies to continuous distributions.
- + Lower Critical Value.
@@ -640,7 +640,7 @@ the Quantile.
- + Upper Critical Value.
@@ -650,7 +650,7 @@ complement of the probability.
- + Survival Function
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/future.html b/doc/sf_and_dist/html/math_toolkit/dist/future.html index a28dbba1c..3f928cefe 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/future.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/future.html @@ -27,7 +27,7 @@ Extras/Future Directions
- + Adding Additional Location and Scale Parameters
@@ -55,7 +55,7 @@ functions.- + An "any_distribution" class
@@ -91,7 +91,7 @@ investigation.- + Higher Level Hypothesis Tests
@@ -111,7 +111,7 @@ expected_mean.- + Integration With Statistical Accumulators
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut.html index 15dcf4614..c8e38ff36 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut.html @@ -36,7 +36,7 @@ are Objects
+
Often you don't want the value of the CDF, but its complement, which
is to say 1-p rather than p.
It is tempting to calculate the CDF and subtract it from 1, but if p
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg.html
index e50e07281..517e26c4e 100644
--- a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg.html
+++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg.html
@@ -87,10 +87,12 @@
Distribution Examples
diff --git a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html index a02997a1a..61ba9d4cb 100644 --- a/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html +++ b/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/weg/cs_eg/chi_sq_intervals.html @@ -139,7 +139,7 @@ _____________________________________________ is between 0.00551 and 0.00729.
negative_binomial_distribution<MyFPType> mydist6(8, 1); // Integer arguments -> MyFPType.
![]() |
+Home | +Libraries | +People | +FAQ | +More | +
+ The scaled-inversed-chi-squared distribution is the conjugate prior distribution + for the variance (σ2) parameter of a normal distribution with known expectation + (μ). As such it has widespread application in Bayesian statistics: +
++ In Bayesian + inference, the strength of belief into certain parameter values + is itself described through a distribution. Parameters hence become themselves + modelled and interpreted as random variables. +
++ In this worked example, we perform such a Bayesian analysis by using + the scaled-inverse-chi-squared distribution as prior and posterior distribution + for the variance parameter of a normal distribution. +
++ For more general information on Bayesian type of analyses, see: +
++ (As the scaled-inversed-chi-squared is another parameterization of the + inverse-gamma distribution, this example could also have used the inverse-gamma + distribution). +
++ Consider precision machines which produce balls for a high-quality ball + bearing. Ideally each ball should have a diameter of precisely 3000 μm + (3 mm). Assume that machines generally produce balls of that size on + average (mean), but individual balls can vary slightly in either direction + following (approximately) a normal distribution. Depending on various + production conditions (e.g. raw material used for balls, workplace temperature + and humidity, maintenance frequency and quality) some machines produce + balls tighter distributed around the target of 3000 μm, while others produce + balls with a wider distribution. Therefore the variance parameter of + the normal distribution of the ball sizes varies from machine to machine. + An extensive survey by the precision machinery manufacturer, however, + has shown that most machines operate with a variance between 15 and 50, + and near 25 μm2 on average. +
++ Using this information, we want to model the variance of the machines. + The variance is strictly positive, and therefore we look for a statistical + distribution with support in the positive domain of the real numbers. + Given the expectation of the normal distribution of the balls is known + (3000 μm), for reasons of conjugacy, it is customary practice in Bayesian + statistics to model the variance to be scaled-inverse-chi-squared distributed. +
++ In a first step, we will try to use the survey information to model the + general knowledge about the variance parameter of machines measured by + the manufacturer. This will provide us with a generic prior distribution + that is applicable if nothing more specific is known about a particular + machine. +
++ In a second step, we will then combine the prior-distribution information + in a Bayesian analysis with data on a specific single machine to derive + a posterior distribution for that machine. +
++ Using the survey results, we try to find the parameter set of a scaled-inverse-chi-squared + distribution so that the properties of this distribution match the results. + Using the mathematical properties of the scaled-inverse-chi-squared distribution + as guideline, we see that that both the mean and mode of the scaled-inverse-chi-squared + distribution are approximately given by the scale parameter (s) of the + distribution. As the survey machines operated at a variance of 25 μm2 on + average, we hence set the scale parameter (sprior) of our prior distribution + equal to this value. Using some trial-and-error and calls to the global + quantile function, we also find that a value of 20 for the degrees-of-freedom + (νprior) parameter is adequate so that most of the prior distribution + mass is located between 15 and 50 (see figure below). +
++ We first construct our prior distribution using these values, and then + list out a few quantiles: +
++ +
+double priorDF = 20.0; +double priorScale = 25.0; + +inverse_chi_squared prior(priorDF, priorScale); +// Using an inverse_gamma distribution instead, we could equivalently write +// inverse_gamma prior(priorDF / 2.0, priorScale * priorDF / 2.0); + +cout << "Prior distribution:" << endl << endl; +cout << " 2.5% quantile: " << quantile(prior, 0.025) << endl; +cout << " 50% quantile: " << quantile(prior, 0.5) << endl; +cout << " 97.5% quantile: " << quantile(prior, 0.975) << endl << endl; + ++
+
++ This produces this output: +
+Prior distribution: + +2.5% quantile: 14.6 +50% quantile: 25.9 +97.5% quantile: 52.1 ++
+ Based on this distribution, we can now calculate the probability of having
+ a machine working with an unusual work precision (variance) at <=
+ 15 or > 50. For this task, we use calls to the boost::math:: functions cdf
+ and complement, respectively,
+ and find a probability of about 0.031 (3.1%) for each case.
+
+ +
+cout << " probability variance <= 15: " << boost::math::cdf(prior, 15.0) << endl; +cout << " probability variance <= 25: " << boost::math::cdf(prior, 25.0) << endl; +cout << " probability variance > 50: " + << boost::math::cdf(boost::math::complement(prior, 50.0)) +<< endl << endl; ++
+
++ This produces this output: +
+probability variance <= 15: 0.031 +probability variance <= 25: 0.458 +probability variance > 50: 0.0318 ++
+ Therefore, only 3.1% of all precision machines produce balls with a variance + of 15 or less (particularly precise machines), but also only 3.2% of + all machines produce balls with a variance of as high as 50 or more (particularly + imprecise machines). Moreover, slightly more than one-half (1 - 0.458 + = 54.2%) of the machines work at a variance greater than 25. +
++ Notice here the distinction between a Bayesian + analysis and a frequentist + analysis: because we model the variance as random variable itself, we + can calculate and straightforwardly interpret probabilities for given + parameter values directly, while such an approach is not possible (and + interpretationally a strict must-not) in the frequentist + world. +
++ In the second step, we investigate a single machine, which is suspected + to suffer from a major fault as the produced balls show fairly high size + variability. Based on the prior distribution of generic machinery performance + (derived above) and data on balls produced by the suspect machine, we + calculate the posterior distribution for that machine and use its properties + for guidance regarding continued machine operation or suspension. +
++ It can be shown that if the prior distribution was chosen to be scaled-inverse-chi-square + distributed, then the posterior distribution is also scaled-inverse-chi-squared-distributed + (prior and posterior distributions are hence conjugate). For more details + regarding conjugacy and formula to derive the parameters set for the + posterior distribution see Conjugate + prior. +
++ Given the prior distribution parameters and sample data (of size n), + the posterior distribution parameters are given by the two expressions: +
++ νposterior = νprior + n +
++ which gives the posteriorDF below, and +
++ sposterior = (νpriorsprior + Σni=1(xi - μ)2) / (νprior + n) +
++ which after some rearrangement gives the formula for the posteriorScale + below. +
++ Machine-specific data consist of 100 balls which were accurately measured + and show the expected mean of 3000 μm and a sample variance of 55 (calculated + for a sample mean defined to be 3000 exactly). From these data, the prior + parameterization, and noting that the term Σni=1(xi - μ)2 equals the sample + variance multiplied by n - 1, it follows that the posterior distribution + of the variance parameter is scaled-inverse-chi-squared distribution + with degrees-of-freedom (νposterior) = 120 and scale (sposterior) = 49.54. +
++ +
+int ballsSampleSize = 100; +cout <<"balls sample size: " << ballsSampleSize << endl; +double ballsSampleVariance = 55.0; +cout <<"balls sample variance: " << ballsSampleVariance << endl; + +double posteriorDF = priorDF + ballsSampleSize; +cout << "prior degrees-of-freedom: " << priorDF << endl; +cout << "posterior degrees-of-freedom: " << posteriorDF << endl; + +double posteriorScale = + (priorDF * priorScale + (ballsSampleVariance * (ballsSampleSize - 1))) / posteriorDF; +cout << "prior scale: " << priorScale << endl; +cout << "posterior scale: " << posteriorScale << endl;+
+
++ An interesting feature here is that one needs only to know a summary + statistics of the sample to parameterize the posterior distribution: + the 100 individual ball measurements are irrelevant, just knowledge of + the sample variance and number of measurements is sufficient. +
++ That produces this output: +
+balls sample size: 100 +balls sample variance: 55 +prior degrees-of-freedom: 20 +posterior degrees-of-freedom: 120 +prior scale: 25 +posterior scale: 49.5 ++
+ To compare the generic machinery performance with our suspect machine, + we calculate again the same quantiles and probabilities as above, and + find a distribution clearly shifted to greater values (see figure). +
+
+
+
+ +
+inverse_chi_squared posterior(posteriorDF, posteriorScale); + + cout << "Posterior distribution:" << endl << endl; + cout << " 2.5% quantile: " << boost::math::quantile(posterior, 0.025) << endl; + cout << " 50% quantile: " << boost::math::quantile(posterior, 0.5) << endl; + cout << " 97.5% quantile: " << boost::math::quantile(posterior, 0.975) << endl << endl; + + cout << " probability variance <= 15: " << boost::math::cdf(posterior, 15.0) << endl; + cout << " probability variance <= 25: " << boost::math::cdf(posterior, 25.0) << endl; + cout << " probability variance > 50: " + << boost::math::cdf(boost::math::complement(posterior, 50.0)) << endl; + ++
+
++ This produces this output: +
+Posterior distribution: + + 2.5% quantile: 39.1 + 50% quantile: 49.8 + 97.5% quantile: 64.9 + + probability variance <= 15: 2.97e-031 + probability variance <= 25: 8.85e-010 + probability variance > 50: 0.489 ++
+ Indeed, the probability that the machine works at a low variance (<= + 15) is almost zero, and even the probability of working at average or + better performance is negligibly small (less than one-millionth of a + permille). On the other hand, with an almost near-half probability (49%), + the machine operates in the extreme high variance range of > 50 characteristic + for poorly performing machines. +
++ Based on this information the operation of the machine is taken out of + use and serviced. +
++ In summary, the Bayesian analysis allowed us to make exact probabilistic + statements about a parameter of interest, and hence provided us results + with straightforward interpretation. +
++ A full sample output is: +
+Inverse_chi_squared_distribution Bayes example: + + Prior distribution: + + 2.5% quantile: 14.6 + 50% quantile: 25.9 + 97.5% quantile: 52.1 + + probability variance <= 15: 0.031 + probability variance <= 25: 0.458 + probability variance > 50: 0.0318 + + balls sample size: 100 + balls sample variance: 55 + prior degrees-of-freedom: 20 + posterior degrees-of-freedom: 120 + prior scale: 25 + posterior scale: 49.5 + Posterior distribution: + + 2.5% quantile: 39.1 + 50% quantile: 49.8 + 97.5% quantile: 64.9 + + probability variance <= 15: 2.97e-031 + probability variance <= 25: 8.85e-010 + probability variance > 50: 0.489 + ++
+ (See also the reference documentation for the Inverse + chi squared Distribution.) +
++ See the full source C++ of this example at ../../../example/inverse_chi_squared_bayes_eg.cpp +
+| + | + |
(See also the reference documentation for the Noncentral Chi Squared Distribution.) @@ -46,7 +46,7 @@