Dear Monks,

Need to calculate the probability that the sum of random variables equals some value.

Training set: Number of objects (e.g. arrays of numbers), calculate the frequency that the sum of the object equals 0, 1, 2..max.

Probability: Find the probability that the sum of a number of random variables equals some number (between 0 and max), given those frequencies from the training set.

Example: After sampling (X) objects, the following frequencies have been found from the training set:
Freq(sum=0) = 0.1
Freq(sum=1) = 0.2
Freq(sum=2) = 0.7

What is the probability that the sum of 2 elements = 4? This code does the job:

#!/usr/bin/perl -w
use strict;
use Algorithm::Loops qw(NestedLoops);
use List::Util qw(sum);

#Set no. of elements, frequencies and the sum value
my $nElements = 2;
my @freq = qw (0.1 0.2 0.7);
my $targetSum = 4;

#Ways of generating sum given no. of elements
my @pos = (0..$#freq);
my (@combo, @matrix);
for my $c (0..$nElements-1) {
    push @combo, [@pos];
    }
my @nest = NestedLoops(\@combo, sub { [ @_ ] } );
for my $n (0..$#nest) {
    push @{$matrix[sum @{$nest[$n]}]}, join ('', @{$nest[$n]});
    }
my @ways = @{$matrix[$targetSum]};

#Now calculate P-value
my @pVal;
foreach my $w (0..$#ways) {
    my @f = split ('', $ways[$w]);
    #print "@f\n";
    my $p = $freq[$f[0]];
    #print $p,"\n";
    for my $g (1..$#f) {
        $p *= $freq[$f[$g]];
        }
    push @pVal, $p;
    }
my $pVal = sum @pVal;
print "P-value(sum of $nElements = $targetSum) = $pVal\n";

#Prints: P-value(sum of 2 = 4) = 0.49

[download]
I have much LARGER SUMS to evaluate that derive from larger number of elements (thousands), with MORE FREQUENCIES (up to 10) and doubt that this code will ever scale up well.

Is there a module that can be used for this ? Or is there some way to do this that would scale up nicely. Convolutions ? FFT ? I am no mathematician, if you may please provide details.

Thank you very much, best.

FFRANK

In reply to Probability sum of random variables by FFRANK

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