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-rw-r--r--lib/Math/SpecFun/Erf.pm565
-rw-r--r--lib/Statistics/Distrib/Normal.pm438
2 files changed, 1003 insertions, 0 deletions
diff --git a/lib/Math/SpecFun/Erf.pm b/lib/Math/SpecFun/Erf.pm
new file mode 100644
index 0000000..c913792
--- /dev/null
+++ b/lib/Math/SpecFun/Erf.pm
@@ -0,0 +1,565 @@
+#
+# Author: Peter J. Acklam
+# Time-stamp: 2000-11-29 23:04:53
+# E-mail: pjacklam@online.no
+# URL: http://home.online.no/~pjacklam
+
+=head1 NAME
+
+Math::SpecFun::Erf - error and scaled and unscaled complementary error
+functions and their inverses
+
+=head1 SYNOPSIS
+
+ use Math::SpecFun::Erf qw(erf erfc erfcx erfinv erfcinv erfcxinv);
+
+Imports all the routines explicitly. Use a subset of the list for the
+routines you want.
+
+ use Math::SpecFun::Erf qw(:all);
+
+Imports all the routines, as well.
+
+=head1 DESCRIPTION
+
+This module implements the error function, C<erf>, and its inverse
+C<erfinv>, the complementary error function, C<erfc>, and its inverse
+C<erfcinv>, and the scaled complementary error function, C<erfcx>, and its
+inverse C<erfcxinv>.
+
+For references and details about the algorithms, see the comments inside
+this module.
+
+=head1 FUNCTIONS
+
+=over 8
+
+=item erf EXPR
+
+=item erf
+
+Returns the error function evaluated at EXPR. If EXPR is omitted, C<$_> is
+used. The error function is
+
+ erf(x) = 2/sqrt(PI) * integral from 0 to x of exp(-t*t) dt
+
+=item erfinv EXPR
+
+=item erfinv
+
+Returns the inverse of the error function evaluated at EXPR. If EXPR is
+omitted, C<$_> is used.
+
+=item erfc EXPR
+
+=item erfc
+
+Returns the complementary error function evaluated at EXPR. If EXPR is
+omitted, C<$_> is used. The complementary error function is
+
+ erfc(x) = 2/sqrt(PI) * integral from x to infinity of exp(-t*t) dt
+ = 1 - erf(x)
+
+Here is a function returning the lower tail probability of the standard
+normal distribution function
+
+ use Math::SpecFun::Erf qw(erfc);
+
+ sub ltpnorm ($) {
+ erfc( - $_[0] / sqrt(2) )/2;
+ }
+
+=item erfcinv EXPR
+
+=item erfcinv
+
+Returns the inverse complementary error function evaluated at EXPR. If EXPR
+is omitted, C<$_> is used.
+
+Here is a function returning the lower tail quantile of the standard normal
+distribution function
+
+ use Math::SpecFun::Erf qw(erfcinv);
+
+ sub ltqnorm ($) {
+ -sqrt(2) * erfcinv( 2 * $_[0] );
+ }
+
+=item erfcx EXPR
+
+=item erfcx
+
+Returns the scaled complementary error function evaluated at EXPR. If EXPR
+is omitted, C<$_> is used. The scaled complementary error function is
+
+ erfcx(x) = exp(x*x) * erfc(x)
+
+=item erfcxinv EXPR
+
+=item erfcxinv
+
+Returns the inverse scaled complementary error function evaluated at EXPR.
+If EXPR is omitted, C<$_> is used.
+
+=back
+
+=head1 HISTORY
+
+=over 4
+
+=item Version 0.03
+
+Added the inverse functions.
+
+=item Version 0.02
+
+Minor code tweaking.
+
+=item Version 0.01
+
+First release.
+
+=back
+
+=head1 AUTHOR
+
+Perl translation by Peter J. Acklam E<lt>pjacklam@online.noE<gt>
+
+FORTRAN code by W. J. Cody, Argonne National Laboratory, March 19, 1990.
+FORTRAN code can be found at http://www.netlib.org/specfun/erf
+
+=head1 COPYRIGHT
+
+Copyright (c) 1999-2000 Peter J. Acklam. All rights reserved.
+This program is free software; you can redistribute it and/or
+modify it under the same terms as Perl itself.
+
+=cut
+
+package Math::SpecFun::Erf;
+require 5.000;
+require Exporter;
+
+use strict;
+use vars qw($VERSION @ISA @EXPORT_OK %EXPORT_TAGS);
+
+$VERSION = '0.02';
+@ISA = qw(Exporter);
+@EXPORT_OK = qw(erf erfc erfcx erfinv erfcinv erfcxinv);
+%EXPORT_TAGS = ( all => [ @EXPORT_OK ] );
+
+########################################################################
+## Internal functions.
+########################################################################
+
+sub calerf {
+ my ($arg, $result, $jint) = @_;
+ local $[ = 1;
+#------------------------------------------------------------------
+#
+# This packet evaluates erf(x), erfc(x), and exp(x*x)*erfc(x)
+# for a real argument x. It contains three FUNCTION type
+# subprograms: ERF, ERFC, and ERFCX (or DERF, DERFC, and DERFCX),
+# and one SUBROUTINE type subprogram, CALERF. The calling
+# statements for the primary entries are:
+#
+# Y=ERF(X) (or Y=DERF(X)),
+#
+# Y=ERFC(X) (or Y=DERFC(X)),
+# and
+# Y=ERFCX(X) (or Y=DERFCX(X)).
+#
+# The routine CALERF is intended for internal packet use only,
+# all computations within the packet being concentrated in this
+# routine. The function subprograms invoke CALERF with the
+# statement
+#
+# CALL CALERF(ARG,RESULT,JINT)
+#
+# where the parameter usage is as follows
+#
+# Function Parameters for CALERF
+# call ARG Result JINT
+#
+# ERF(ARG) ANY REAL ARGUMENT ERF(ARG) 0
+# ERFC(ARG) ABS(ARG) < XBIG ERFC(ARG) 1
+# ERFCX(ARG) XNEG < ARG < XMAX ERFCX(ARG) 2
+#
+# The main computation evaluates near-minimax approximations
+# from "Rational Chebyshev approximations for the error function"
+# by W. J. Cody, Math. Comp., 1969, PP. 631-638. This
+# transportable program uses rational functions that theoretically
+# approximate erf(x) and erfc(x) to at least 18 significant
+# decimal digits. The accuracy achieved depends on the arithmetic
+# system, the compiler, the intrinsic functions, and proper
+# selection of the machine-dependent constants.
+#
+#*******************************************************************
+#*******************************************************************
+#
+# Explanation of machine-dependent constants
+#
+# XMIN = the smallest positive floating-point number.
+# XINF = the largest positive finite floating-point number.
+# XNEG = the largest negative argument acceptable to ERFCX;
+# the negative of the solution to the equation
+# 2*exp(x*x) = XINF.
+# XSMALL = argument below which erf(x) may be represented by
+# 2*x/sqrt(pi) and above which x*x will not underflow.
+# A conservative value is the largest machine number X
+# such that 1.0 + X = 1.0 to machine precision.
+# XBIG = largest argument acceptable to ERFC; solution to
+# the equation: W(x) * (1-0.5/x**2) = XMIN, where
+# W(x) = exp(-x*x)/[x*sqrt(pi)].
+# XHUGE = argument above which 1.0 - 1/(2*x*x) = 1.0 to
+# machine precision. A conservative value is
+# 1/[2*sqrt(XSMALL)]
+# XMAX = largest acceptable argument to ERFCX; the minimum
+# of XINF and 1/[sqrt(pi)*XMIN].
+#
+# Approximate values for some important machines are:
+#
+# XMIN XINF XNEG XSMALL
+#
+# CDC 7600 (S.P.) 3.13E-294 1.26E+322 -27.220 7.11E-15
+# CRAY-1 (S.P.) 4.58E-2467 5.45E+2465 -75.345 7.11E-15
+# IEEE (IBM/XT,
+# SUN, etc.) (S.P.) 1.18E-38 3.40E+38 -9.382 5.96E-8
+# IEEE (IBM/XT,
+# SUN, etc.) (D.P.) 2.23D-308 1.79D+308 -26.628 1.11D-16
+# IBM 195 (D.P.) 5.40D-79 7.23E+75 -13.190 1.39D-17
+# UNIVAC 1108 (D.P.) 2.78D-309 8.98D+307 -26.615 1.73D-18
+# VAX D-Format (D.P.) 2.94D-39 1.70D+38 -9.345 1.39D-17
+# VAX G-Format (D.P.) 5.56D-309 8.98D+307 -26.615 1.11D-16
+#
+#
+# XBIG XHUGE XMAX
+#
+# CDC 7600 (S.P.) 25.922 8.39E+6 1.80X+293
+# CRAY-1 (S.P.) 75.326 8.39E+6 5.45E+2465
+# IEEE (IBM/XT,
+# SUN, etc.) (S.P.) 9.194 2.90E+3 4.79E+37
+# IEEE (IBM/XT,
+# SUN, etc.) (D.P.) 26.543 6.71D+7 2.53D+307
+# IBM 195 (D.P.) 13.306 1.90D+8 7.23E+75
+# UNIVAC 1108 (D.P.) 26.582 5.37D+8 8.98D+307
+# VAX D-Format (D.P.) 9.269 1.90D+8 1.70D+38
+# VAX G-Format (D.P.) 26.569 6.71D+7 8.98D+307
+#
+#*******************************************************************
+#*******************************************************************
+#
+# Error returns
+#
+# The program returns ERFC = 0 for ARG >= XBIG;
+#
+# ERFCX = XINF for ARG < XNEG;
+# and
+# ERFCX = 0 for ARG >= XMAX.
+#
+#
+# Intrinsic functions required are:
+#
+# ABS, AINT, EXP
+#
+#
+# Author: W. J. Cody
+# Mathematics and Computer Science Division
+# Argonne National Laboratory
+# Argonne, IL 60439
+#
+# Latest modification: March 19, 1990
+#
+# Translation to Perl by Peter J. Acklam, December 3, 1999
+#
+#------------------------------------------------------------------
+ my ($i);
+ my ($x, $del, $xden, $xnum, $y, $ysq);
+#------------------------------------------------------------------
+# Mathematical constants
+#------------------------------------------------------------------
+ my ($four, $one, $half, $two, $zero) = (4, 1, 0.5, 2, 0);
+ my $sqrpi = 5.6418958354775628695e-1;
+ my $thresh = 0.46875;
+ my $sixten = 16;
+#------------------------------------------------------------------
+# Machine-dependent constants
+#------------------------------------------------------------------
+ my ($xinf, $xneg, $xsmall) = (1.79e308, -26.628, 1.11e-16);
+ my ($xbig, $xhuge, $xmax) = (26.543, 6.71e7, 2.53e307);
+#------------------------------------------------------------------
+# Coefficients for approximation to erf in first interval
+#------------------------------------------------------------------
+ my @a = (3.16112374387056560e00, 1.13864154151050156e02,
+ 3.77485237685302021e02, 3.20937758913846947e03,
+ 1.85777706184603153e-1);
+ my @b = (2.36012909523441209e01, 2.44024637934444173e02,
+ 1.28261652607737228e03, 2.84423683343917062e03);
+#------------------------------------------------------------------
+# Coefficients for approximation to erfc in second interval
+#------------------------------------------------------------------
+ my @c = (5.64188496988670089e-1, 8.88314979438837594e0,
+ 6.61191906371416295e01, 2.98635138197400131e02,
+ 8.81952221241769090e02, 1.71204761263407058e03,
+ 2.05107837782607147e03, 1.23033935479799725e03,
+ 2.15311535474403846e-8);
+ my @d = (1.57449261107098347e01, 1.17693950891312499e02,
+ 5.37181101862009858e02, 1.62138957456669019e03,
+ 3.29079923573345963e03, 4.36261909014324716e03,
+ 3.43936767414372164e03, 1.23033935480374942e03);
+#------------------------------------------------------------------
+# Coefficients for approximation to erfc in third interval
+#------------------------------------------------------------------
+ my @p = (3.05326634961232344e-1, 3.60344899949804439e-1,
+ 1.25781726111229246e-1, 1.60837851487422766e-2,
+ 6.58749161529837803e-4, 1.63153871373020978e-2);
+ my @q = (2.56852019228982242e00, 1.87295284992346047e00,
+ 5.27905102951428412e-1, 6.05183413124413191e-2,
+ 2.33520497626869185e-3);
+#------------------------------------------------------------------
+ $x = $arg;
+ $y = abs($x);
+ if ($y <= $thresh) {
+#------------------------------------------------------------------
+# Evaluate erf for |X| <= 0.46875
+#------------------------------------------------------------------
+ $ysq = $zero;
+ if ($y > $xsmall) { $ysq = $y * $y }
+ $xnum = $a[5]*$ysq;
+ $xden = $ysq;
+ for (my $i = 1 ; $i <= 3 ; ++$i) {
+ $xnum = ($xnum + $a[$i]) * $ysq;
+ $xden = ($xden + $b[$i]) * $ysq;
+ }
+ $$result = $x * ($xnum + $a[4]) / ($xden + $b[4]);
+ if ($jint != 0) { $$result = $one - $$result }
+ if ($jint == 2) { $$result = exp($ysq) * $$result }
+ goto x800;
+#------------------------------------------------------------------
+# Evaluate erfc for 0.46875 <= |X| <= 4.0
+#------------------------------------------------------------------
+ } elsif ($y <= $four) {
+ $xnum = $c[9]*$y;
+ $xden = $y;
+ for (my $i = 1 ; $i <= 7 ; ++$i) {
+ $xnum = ($xnum + $c[$i]) * $y;
+ $xden = ($xden + $d[$i]) * $y;
+ }
+ $$result = ($xnum + $c[8]) / ($xden + $d[8]);
+ if ($jint != 2) {
+ $ysq = int($y*$sixten)/$sixten;
+ $del = ($y-$ysq)*($y+$ysq);
+ $$result = exp(-$ysq*$ysq) * exp(-$del) * $$result;
+ }
+#------------------------------------------------------------------
+# Evaluate erfc for |X| > 4.0
+#------------------------------------------------------------------
+ } else {
+ $$result = $zero;
+ if ($y >= $xbig) {
+ if (($jint != 2) || ($y >= $xmax)) { goto x300 }
+ if ($y >= $xhuge) {
+ $$result = $sqrpi / $y;
+ goto x300;
+ }
+ }
+ $ysq = $one / ($y * $y);
+ $xnum = $p[6]*$ysq;
+ $xden = $ysq;
+ for (my $i = 1 ; $i <= 4 ; ++$i) {
+ $xnum = ($xnum + $p[$i]) * $ysq;
+ $xden = ($xden + $q[$i]) * $ysq;
+ }
+ $$result = $ysq *($xnum + $p[5]) / ($xden + $q[5]);
+ $$result = ($sqrpi - $$result) / $y;
+ if ($jint != 2) {
+ $ysq = int($y*$sixten)/$sixten;
+ $del = ($y-$ysq)*($y+$ysq);
+ $$result = exp(-$ysq*$ysq) * exp(-$del) * $$result;
+ }
+ }
+#------------------------------------------------------------------
+# Fix up for negative argument, erf, etc.
+#------------------------------------------------------------------
+ x300:
+ if ($jint == 0) {
+ $$result = ($half - $$result) + $half;
+ if ($x < $zero) { $$result = -$$result }
+ } elsif ($jint == 1) {
+ if ($x < $zero) { $$result = $two - $$result }
+ } else {
+ if ($x < $zero) {
+ if ($x < $xneg) {
+ $$result = $xinf;
+ } else {
+ $ysq = int($x*$sixten)/$sixten;
+ $del = ($x-$ysq)*($x+$ysq);
+ $y = exp($ysq*$ysq) * exp($del);
+ $$result = ($y+$y) - $$result;
+ }
+ }
+ }
+ x800:
+ return 1;
+#---------- Last card of CALERF ----------
+}
+
+sub erf {
+ my $x = @_ ? $_[0] : $_;
+#--------------------------------------------------------------------
+#
+# This subprogram computes approximate values for erf(x).
+# (see comments heading CALERF).
+#
+# Author/date: W. J. Cody, January 8, 1985
+#
+# Translation to Perl by Peter J. Acklam, December 3, 1999
+#
+#--------------------------------------------------------------------
+ my $result;
+ my $jint = 0;
+ calerf($x, \$result, $jint);
+ my $erf = $result;
+ return $erf;
+#---------- Last card of ERF ----------
+}
+
+########################################################################
+## User functions.
+########################################################################
+
+sub erfc {
+ my $x = @_ ? $_[0] : $_;
+#--------------------------------------------------------------------
+#
+# This subprogram computes approximate values for erfc(x).
+# (see comments heading CALERF).
+#
+# Author/date: W. J. Cody, January 8, 1985
+#
+# Translation to Perl by Peter J. Acklam, December 3, 1999
+#
+#--------------------------------------------------------------------
+ my ($result);
+ my $jint = 1;
+ calerf($x, \$result, $jint);
+ my $erfc = $result;
+ return $erfc;
+#---------- Last card of ERFC ----------
+}
+
+sub erfcx {
+ my $x = @_ ? $_[0] : $_;
+#------------------------------------------------------------------
+#
+# This subprogram computes approximate values for exp(x*x) * erfc(x).
+# (see comments heading CALERF).
+#
+# Author/date: W. J. Cody, March 30, 1987
+#
+# Translation to Perl by Peter J. Acklam, December 3, 1999
+#
+#------------------------------------------------------------------
+ my ($result);
+ my $jint = 2;
+ calerf($x, \$result, $jint);
+ my $erfcx = $result;
+ return $erfcx;
+#---------- Last card of ERFCX ----------
+}
+
+sub erfinv {
+ my $y = @_ ? $_[0] : $_;
+
+ return 0 if $y == 0;
+ return erfcinv(1-$y) if $y > 0.5;
+ return -erfcinv(1+$y) if $y < -0.5;
+
+ #
+ # Halley's rational 3rd order method:
+ # u <- f(x)/f'(x)
+ # v <- f''(x)/f'(x)
+ # x <- x - u/(1-u*v/2)
+ #
+ # Here:
+ # f(x) = erf(x) - y
+ # f'(x) = 2/sqrt(pi)*exp(-x*x)
+ # f''(x) = -4/sqrt(pi)*x*exp(-x*x)
+ #
+ my $x = 0;
+ my $dx;
+ my $c = .88622692545275801364908374167055; # sqrt(pi)/2
+ my $eps = 5e-15;
+ do {
+ my $f = erf($x) - $y;
+ my $u = $c*$f*exp($x*$x);
+ $dx = -$u/(1+$u*$x);
+ $x += $dx;
+ } until abs($dx/$x) <= $eps;
+ return $x;
+}
+
+sub erfcinv {
+ my $y = @_ ? $_[0] : $_;
+
+ return 0 if $y == 1;
+
+ my $flag = $y > 1;
+ $y = 2 - $y if $flag;
+
+ #
+ # Halley's rational 3rd order method:
+ # u <- f(x)/f'(x)
+ # v <- f''(x)/f'(x)
+ # x <- x - u/(1-u*v/2)
+ #
+ # Here:
+ # f(x) = erfc(x) - y
+ # f'(x) = -2/sqrt(pi)*exp(-x*x)
+ # f''(x) = 4/sqrt(pi)*x*exp(-x*x)
+ #
+ my $x = 0;
+ my $dx;
+ my $c = -.88622692545275801364908374167055; # sqrt(pi)/2
+ my $eps = 5e-15;
+ do {
+ my $u = $c*(erfcx($x) - $y*exp($x*$x));
+ $dx = -$u/(1+$u*$x);
+ $x += $dx;
+ } until abs($dx/$x) <= $eps;
+
+ return $flag ? -$x : $x;
+}
+
+sub erfcxinv {
+ my $y = @_ ? $_[0] : $_;
+
+ return 0 if $y == 1;
+
+ #
+ # Halley's 3rd order method:
+ # u <- f(x)/f'(x)
+ # v <- f''(x)/f'(x)
+ # x <- x - u/(1-u*v/2)
+ #
+ # Here:
+ # f(x) = erfcx(x) - y
+ # f'(x) = 2*(x*erfcx(x)-1/sqrt(pi));
+ # f''(x) = (2+4*x*x)*erfcx(x) - 4*x/sqrt(pi);
+ #
+ my $x = 0;
+ my $dx;
+ my $c = .56418958354775628694807945156079; # 1/sqrt(pi)
+ my $d = 2.2567583341910251477923178062432; # 4/sqrt(pi)
+ my $eps = 5e-15;
+ do {
+ my $f = erfcx($x) - $y;
+ my $df = 2*($x*erfcx($x)-$c);
+ my $ddf = (2+4*$x*$x)*erfcx($x) - $x*$d;
+ my $u = $f/$df;
+ my $v = $ddf/$df;
+ $dx = -$u/(1-$u*$v/2);
+ $x += $dx;
+ } until abs($dx/$x) <= $eps;
+ return $x;
+}
diff --git a/lib/Statistics/Distrib/Normal.pm b/lib/Statistics/Distrib/Normal.pm
new file mode 100644
index 0000000..58cf737
--- /dev/null
+++ b/lib/Statistics/Distrib/Normal.pm
@@ -0,0 +1,438 @@
+#
+# Author: Peter J. Acklam
+# Time-stamp: 2000-06-02 23:42:57
+# E-mail: pjacklam@online.no
+# URL: http://home.online.no/~pjacklam
+
+=head1 NAME
+
+Statistics::Distrib::Normal - the normal distribution
+
+=head1 SYNOPSIS
+
+ use Statistics::Distrib::Normal;
+
+ $dist = new Statistics::Distrib::Normal;
+
+ $dist->mu(3); # set the location parameter
+ $dist->sigma(5); # set the scale parameter
+ @x = $dist->rand(10); # generate random numbers
+
+ # or
+
+ @x = Statistics::Distrib::Normal->new(Mu => 3, Sigma => 5)->rand(10);
+
+=head1 DESCRIPTION
+
+This module contains miscellaneous functions related to the normal
+distribution.
+
+=cut
+
+package Statistics::Distrib::Normal;
+require 5.000;
+
+use strict;
+use vars qw($VERSION);
+use Carp;
+
+$VERSION = '0.01';
+
+use constant PI => 4 * atan2 1, 1;
+use constant TWOPI => 2 * PI;
+
+# the smallest positive floating-point number such that 1+EPS > 1
+use constant EPS => 2.220446049250313080847263336181640625e-016;
+
+##
+## Constructor
+##
+
+=head1 CONSTRUCTOR
+
+=over 4
+
+=item new ( [ OPTIONS ] )
+
+C<OPTIONS> is a list of options given in the form of key-value
+pairs, just like a hash table. Valid options are
+
+=over 8
+
+=item B<Mu>
+
+Sets the mu parameter (the mean) of the distribution to the specified
+value.
+
+=item B<Sigma>
+
+Sets the sigma parameter (the standard deviation) of the distribution
+to the specified value.
+
+=back
+
+=back
+
+=cut
+
+sub new {
+ my $self = shift;
+ my $class = ref($self) || $self;
+ my %arg = @_;
+
+ my %hash = ( mu => 0,
+ sigma => 1,
+ );
+
+ my $me = bless \%hash, $class;
+
+ foreach my $key ( keys %arg ) {
+ $me->mu($arg{Mu}), next if $key eq 'Mu';
+ $me->sigma($arg{Sigma}), next if $key eq 'Sigma';
+ carp "Unknown option $arg{$key} ignored";
+ }
+
+ return $me;
+}
+
+##
+## Methods
+##
+
+=pod
+
+=head1 METHODS
+
+=over 4
+
+=item mean ( [ MEAN ] )
+
+Set the mu parameter (the mean) of the distribution to C<MU>. If
+C<MU> is omitted, the current value of mu is returned.
+
+=cut
+
+sub mu {
+ my $me = shift;
+ croak 'Too many arguments' if @_ > 1;
+ if ( @_ ) {
+ $me->{mu} = shift;
+ return 1;
+ }
+ return $me->{mu};
+}
+
+=pod
+
+=item sigma ( [ SDEV ] )
+
+Set the sigma parameter (the standard deviation) of the distribution
+to C<SIGMA>. If C<SIGMA> is omitted, the current value of sigma is
+returned.
+
+=cut
+
+sub sigma {
+ my $me = shift;
+ croak 'Too many arguments' if @_ > 1;
+ if ( @_ ) {
+ my $sigma = shift;
+ croak 'Standard deviation most be positive' unless $sigma > 0;
+ $me->{sigma} = $sigma;
+ return 1;
+ }
+ return $me->{sigma};
+}
+
+=pod
+
+=item pdf ( X1 [, X2 [, X3 ... ] ] )
+
+Evaluate the probability density function at C<X1>, C<X2>, C<X3>, ...
+
+=cut
+
+sub pdf {
+ my $me = shift;
+ croak 'Not enough arguments' unless @_;
+ my $mu = $me->{mu};
+ my $sigma = $me->{sigma};
+ my $const = log(TWOPI * $sigma * $sigma);
+ my @f;
+ foreach my $x ( @_ ) {
+ my $z = ( $x - $mu ) / $sigma;
+ push @f, exp( -0.5 * ( $const + $z*$z ) );
+ }
+ return @f;
+}
+
+=pod
+
+=item ltp ( X1 [, X2 [, ... ] ] )
+
+Evaluate the lower tail probability function at C<X1>, C<X2>, C<X3>,
+...
+
+=cut
+
+sub ltp {
+ my $me = shift;
+ croak 'Not enough arguments' unless @_;
+ my $mu = $me->{mu};
+ my $sigma = $me->{sigma};
+
+ require Math::SpecFun::Erf;
+ import Math::SpecFun::Erf qw(erfc);
+
+ my @p;
+ foreach my $x ( @_ ) {
+ my $z = ( $x - $mu ) / $sigma;
+ push @p, erfc( - $_[0] / sqrt(2) )/2;
+ }
+ return @p;
+}
+
+=pod
+
+=item utp ( X1 [, X2 [, ... ] ] )
+
+Evaluate the upper tail probability function at C<X1>, C<X2>, C<X3>,
+...
+
+=cut
+
+sub utp {
+ my $me = shift;
+ croak 'Not enough arguments' unless @_;
+ my $mu = $me->{mu};
+ my $sigma = $me->{sigma};
+
+ require Math::SpecFun::Erf;
+ import Math::SpecFun::Erf qw(erfc);
+
+ my @p;
+ foreach my $x ( @_ ) {
+ my $z = ( $x - $mu ) / $sigma;
+ push @p, erfc( $_[0] / sqrt(2) )/2;
+ }
+ return @p;
+}
+
+=pod
+
+=item ltq ( P1 [, P2 [, ... ] ] )
+
+Returns the lower tail quantile for the probabilities C<P1>, C<P2>,
+C<P3>, ...
+
+=cut
+
+sub ltq {
+ croak 'Method not implemented yet';
+}
+
+=pod
+
+=item utq ( P1 [, P2 [, P3 ... ] ] )
+
+Returns the upper tail quantile for the probabilities C<P1>, C<P2>,
+C<P3>, ...
+
+=cut
+
+sub utq {
+ croak 'Method not implemented yet';
+}
+
+=pod
+
+=item intprob( XLOW, XHIGH )
+
+Interval probability. Returns the probability of an outcome in the
+interval from XLOW to XHIGH.
+
+=cut
+
+sub intprob {
+ my $me = shift;
+ croak 'Bad number of arguments' unless @_ == 2;
+ my ($xlow, $xhigh) = @_;
+ return 0 unless $xlow < $xhigh;
+ my $mu = $me->{mu};
+ my $sigma = $me->{sigma};
+
+ if ( $mu < $xlow ) {
+ return $me->utp($xlow) - $me->utp($xhigh);
+ } else {
+ return $me->ltp($xhigh) - $me->ltp($xlow);
+ }
+}
+
+=pod
+
+=item rand( [ NUM ] )
+
+Generate random variables. If C<NUM> is omitted, a single variable is
+returned.
+
+=cut
+
+sub rand {
+ my $me = shift;
+ my $num;
+ if ( @_ ) {
+ $num = shift;
+ croak 'Too many arguments' if @_;
+ croak 'Argument must be positive integer'
+ unless ($num == int $num) && ($num > 0);
+ } else {
+ $num = 1;
+ }
+
+ # Generate the random variables by the Box-Muller method.
+ my @z;
+ my $mu = $me->{mu};
+ my $sigma = $me->{sigma};
+ my $const = -2 * $sigma * $sigma;
+ my $i;
+ for ( $i = 0 ; $i < $num ; $i += 2 ) {
+ my $r = sqrt $const * log rand;
+ my $t = TWOPI * rand;
+ push @z, $mu + $r * sin $t, $mu + $r * cos $t;
+ }
+ pop @z if $i > $num;
+ return @z;
+}
+
+=pod
+
+=item expectation ()
+
+Return the expectation of the distribution.
+
+=cut
+
+sub expectation {
+ my $me = shift;
+ croak 'Too many arguments' if @_;
+ return $me->{mu};
+}
+
+=pod
+
+=item variance ()
+
+Return the variance of the distribution.
+
+=cut
+
+sub variance {
+ my $me = shift;
+ croak 'Too many arguments' if @_;
+ return $me->{sigma}**2;
+}
+
+=pod
+
+=item skewness ()
+
+Return the skewness of the distribution.
+
+=cut
+
+sub skewness {
+ my $me = shift;
+ croak 'Too many arguments' if @_;
+ return 0;
+}
+
+=pod
+
+=item kurtosis ()
+
+Return the kurtosis (normalized) of the distribution.
+
+=cut
+
+sub kurtosis {
+ my $me = shift;
+ croak 'Too many arguments' if @_;
+ return 0;
+}
+
+=item dmo
+
+Direct moments for the distribution.
+
+Not implemented yet.
+
+=cut
+
+sub dmo {
+ croak 'Method not implemented yet';
+}
+
+=pod
+
+=item cmo
+
+Central moments for the distribution.
+
+=cut
+
+sub cmo {
+ croak 'Method not implemented yet';
+}
+
+=pod
+
+=item mode ()
+
+Returns the mode of the distribution.
+
+=cut
+
+sub mode {
+ my $me = shift;
+ croak 'Too many input arguments' if @_;
+ return $me->{mu};
+}
+
+=back
+
+=head1 BUGS
+
+None known.
+
+=head1 LIMITATIONS
+
+Degenerate cases are not allowed for most methods; e.g., a
+distribution with zero variance.
+
+=head1 HISTORY
+
+=over 4
+
+=item Version 0.02
+
+Code formatting changes.
+
+=item Version 0.01
+
+First release.
+
+=back
+
+=head1 AUTHOR
+
+Peter J. Acklam E<lt>pjacklam@online.noE<gt>.
+
+=head1 COPYRIGHT/LICENSE
+
+Copyright (c) 1999-2000 Peter J. Acklam. All rights reserved.
+This program is free software; you can redistribute it and/or
+modify it under the same terms as Perl itself.
+
+=cut
+
+1; # Modules must return true.
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