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| 1 : | maria | 42 | #!/usr/local/bin/webwork-perl |
| 2 : | |||
| 3 : | maria | 55 | sub _PGstatisticsmacros_init { |
| 4 : | foreach my $t (@Distributions::EXPORT_OK) { | ||
| 5 : | *{$t} = *{"Distributions::$t"} | ||
| 6 : | } | ||
| 7 : | } | ||
| 8 : | |||
| 9 : | maria | 52 | =head1 Statistics Macros |
| 10 : | |||
| 11 : | maria | 42 | =head3 Normal distribution |
| 12 : | |||
| 13 : | =pod | ||
| 14 : | |||
| 15 : | maria | 63 | Usage: normal_prob(a, b, mean=>0, deviation=>1); |
| 16 : | maria | 42 | |
| 17 : | Computes the probability of x being in the interval (a,b) for normal distribution. | ||
| 18 : | maria | 52 | The first two arguments are required. Use '-infty' for negative infinity, and 'infty' or '+infty' for positive infinity. |
| 19 : | The mean and deviation are optional, and are 0 and 1 respectively by default. | ||
| 20 : | maria | 55 | Load PGnumericalmacros.pl in your problem if you use this method. |
| 21 : | maria | 42 | |
| 22 : | =cut | ||
| 23 : | |||
| 24 : | maria | 63 | sub normal_prob { |
| 25 : | maria | 55 | warn 'You must also load PGnumericalmacros to use PGstatisticsmacros' unless defined(&_PGnumericalmacros_init); |
| 26 : | |||
| 27 : | maria | 42 | my $a = shift; |
| 28 : | maria | 55 | my $b = shift; |
| 29 : | my %options=@_; | ||
| 30 : | |||
| 31 : | my $mean = $options{'mean'} if defined ($options{'mean'}); | ||
| 32 : | maria | 42 | $mean = 0 unless defined $mean; |
| 33 : | maria | 55 | |
| 34 : | my $deviation = $options{'deviation'} if defined ($options{'deviation'}); | ||
| 35 : | $deviation = 1 unless defined $deviation; | ||
| 36 : | |||
| 37 : | maria | 42 | if ($deviation <= 0) { |
| 38 : | warn 'Deviation must be a positive number.'; | ||
| 39 : | return; | ||
| 40 : | } | ||
| 41 : | maria | 55 | |
| 42 : | my $z_score_of_a; | ||
| 43 : | my $z_score_of_b; | ||
| 44 : | |||
| 45 : | if ( $a eq '-infty' ) { | ||
| 46 : | $z_score_of_a = -6; | ||
| 47 : | } else { | ||
| 48 : | $z_score_of_a = ($a - $mean)/$deviation; | ||
| 49 : | } | ||
| 50 : | |||
| 51 : | if (($b eq 'infty') or ($b eq '+infty')) { | ||
| 52 : | $z_score_of_b = 6; | ||
| 53 : | } else { | ||
| 54 : | $z_score_of_b = ($b - $mean)/$deviation; | ||
| 55 : | } | ||
| 56 : | |||
| 57 : | maria | 42 | my $function = sub { my $x=shift; |
| 58 : | $E**(-$x**2/2)/sqrt(2*$PI); | ||
| 59 : | }; | ||
| 60 : | maria | 55 | |
| 61 : | maria | 42 | my $prob = romberg($function, $z_score_of_a, $z_score_of_b); |
| 62 : | $prob; | ||
| 63 : | } | ||
| 64 : | |||
| 65 : | =head3 "Inverse" of normal distribution | ||
| 66 : | |||
| 67 : | =pod | ||
| 68 : | |||
| 69 : | maria | 63 | Usage: normal_dist(prob, mean=>0, deviation=>1); |
| 70 : | maria | 42 | |
| 71 : | Computes the positive number b such that the probability of x being in the interval (0,b) | ||
| 72 : | is equal to the given probability (first argument). The mean and deviation are | ||
| 73 : | optional, and are 0 and 1 respectively by default. | ||
| 74 : | maria | 52 | Caution: since students may use tables, they may only be able to provide the answer correct to 2 or 3 |
| 75 : | decimal places. Use tolerance when evaluating answers. | ||
| 76 : | maria | 55 | Load PGnumericalmacros.pl if you use this method. |
| 77 : | maria | 42 | |
| 78 : | =cut | ||
| 79 : | |||
| 80 : | maria | 63 | sub normal_dist { |
| 81 : | maria | 55 | warn 'You must also load PGnumericalmacros to use PGstatisticsmacros' unless defined(&_PGnumericalmacros_init); |
| 82 : | |||
| 83 : | my $prob = shift; | ||
| 84 : | my %options=@_; | ||
| 85 : | |||
| 86 : | maria | 42 | my $mean = $options{'mean'} if defined ($options{'mean'}); |
| 87 : | $mean = 0 unless defined $mean; | ||
| 88 : | maria | 55 | |
| 89 : | maria | 42 | my $deviation = $options{'deviation'} if defined ($options{'deviation'}); |
| 90 : | $deviation = 1 unless defined $deviation; | ||
| 91 : | maria | 55 | |
| 92 : | maria | 42 | if ($deviation <= 0) { |
| 93 : | warn 'Deviation must be a positive number.'; | ||
| 94 : | return; | ||
| 95 : | } | ||
| 96 : | maria | 55 | |
| 97 : | maria | 42 | my $function = sub { my $x=shift; |
| 98 : | $E**(-$x**2/2)/sqrt(2*$PI); | ||
| 99 : | }; | ||
| 100 : | maria | 55 | |
| 101 : | maria | 42 | my $z_score_of_b = inv_romberg($function, 0, $prob); |
| 102 : | maria | 55 | |
| 103 : | maria | 42 | my $b = $z_score_of_b * $deviation + $mean; |
| 104 : | $b; | ||
| 105 : | } | ||
| 106 : | |||
| 107 : | ########################################## | ||
| 108 : | |||
| 109 : | 1; | ||
| 110 : | maria | 63 |
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