View of /trunk/webwork/system/courseScripts/PGstatisticsmacros.pl

    1 #!/usr/local/bin/webwork-perl
    2 
    3 =head1 Statistics Macros
    4 
    5 =head3 Normal distribution
    6 
    7 =pod
    8 
    9   Usage: normal(a, b, mean=>0, deviation=>1);
   10 
   11 Computes the probability of x being in the interval (a,b) for normal distribution.
   12 The first two arguments are required. Use '-infty' for negative infinity, and 'infty' or '+infty' for positive infinity.
   13 The mean and deviation are optional, and are 0 and 1 respectively by default.
   14 
   15 =cut
   16 
   17 sub normal {
   18   my $a = shift;
   19   my $b = shift;
   20         my %options = @_;
   21         my $mean = $options{'mean'} if defined ($options{'mean'});
   22         $mean = 0 unless defined $mean;
   23         my $deviation = $options{'deviation'} if defined ($options{'deviation'});
   24   $deviation = 1 unless defined $deviation;
   25   if ($deviation <= 0) {
   26     warn 'Deviation must be a positive number.';
   27     return;
   28     }
   29   if ( $a eq '-infty' ) { $a = -6*$deviation + $mean; }
   30         if ( $b eq 'infty' ) { $b = 6*$deviation + $mean; }
   31   if ( $b eq '+infty' ) { $b = 6*$deviation + $mean; }
   32         my $z_score_of_a = ($a - $mean)/$deviation;
   33         my $z_score_of_b = ($b - $mean)/$deviation;
   34         my $function = sub { my $x=shift;
   35                              $E**(-$x**2/2)/sqrt(2*$PI);
   36                              };
   37         my $prob = romberg($function, $z_score_of_a, $z_score_of_b);
   38         $prob;
   39 }
   40 
   41 =head3 "Inverse" of normal distribution
   42 
   43 =pod
   44 
   45   Usage: inv_normal(prob, mean=>0, deviation=>1);
   46 
   47 Computes the positive number b such that the probability of x being in the interval (0,b)
   48 is equal to the given probability (first argument). The mean and deviation are
   49 optional, and are 0 and 1 respectively by default.
   50 Caution: since students may use tables, they may only be able to provide the answer correct to 2 or 3
   51 decimal places. Use tolerance when evaluating answers.
   52 
   53 =cut
   54 
   55 sub inv_normal {
   56   my $prob = shift;
   57   my %options = @_;
   58         my $mean = $options{'mean'} if defined ($options{'mean'});
   59         $mean = 0 unless defined $mean;
   60         my $deviation = $options{'deviation'} if defined ($options{'deviation'});
   61         $deviation = 1 unless defined $deviation;
   62         if ($deviation <= 0) {
   63                 warn 'Deviation must be a positive number.';
   64                 return;
   65                 }
   66         my $function = sub { my $x=shift;
   67                              $E**(-$x**2/2)/sqrt(2*$PI);
   68                              };
   69         my $z_score_of_b = inv_romberg($function, 0, $prob);
   70         my $b = $z_score_of_b * $deviation + $mean;
   71         $b;
   72 }
   73 
   74 ##########################################
   75 
   76 1;