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+#
+# 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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