$ ./ana_2.pl 16 4 65536
N=65536, V=4, X=65536 integral(65536)=139415.765849051, integral(0)=136533.333333333
Expected collisions: 2882.43251571807

$ ./ana_2.pl 16 4 16384
N=65536, V=4, X=16384 integral(16384)=136541.854969116, integral(0)=136533.333333333
Expected collisions: 8.52163578287582

$ ./ana_2.pl 14 3 16384
N=16384, V=3, X=16384 integral(16384)=31411.9141476821, integral(0)=30037.3333333333
Expected collisions: 1374.58081434877

$ ./ana_2.pl 16 10 32768
N=65536, V=10, X=32768 integral(32768)=191953.190301726, integral(0)=191952.863492063
Expected collisions: 0.326809662627056

##</code><code>##

$ cat ana_2.pl
#!/usr/bin/perl
#
#   ana_2.pl N V X
#
#   N=vector size (bits), V=number of vectors, X=sample number
#
use strict;
use warnings;
use feature ':5.10';

my $n=shift; $n = 1<<$n;
my $v=shift;
my $x=shift;

my ($exp1, $exp2, $exp3);

given ($v) {
    when ( 1) { $exp1=ex_1($n, $x), $exp2=ex_1($n, 0) }
    when ( 2) { $exp1=ex_2($n, $x), $exp2=ex_2($n, 0) }
    when ( 3) { $exp1=ex_3($n, $x), $exp2=ex_3($n, 0) }
    when ( 4) { $exp1=ex_4($n, $x), $exp2=ex_4($n, 0) }
    when (10) { $exp1=ex_10($n, $x), $exp2=ex_10($n, 0) }
    default  { die "Need symbolic integral form for $v vectors!\n"; }
}

$exp3 = $exp1-$exp2;

print "N=$n, V=$v, X=$x integral($x)=$exp1, integral(0)=$exp2\n";
print "Expected collisions: $exp3\n";

sub ex_1 {
    my ($N, $X) = @_;
    return      $N  *exp(  -$X/$N)
            + $X;
}

sub ex_2 {
    my ($N, $X) = @_;
    return     -$N/2*exp(-2*$X/$N)
            + 2*$N  *exp(  -$X/$N)
            + $X;
}

sub ex_3 {
    my ($N, $X) = @_;
    return      $N/3*exp(-3*$X/$N)
            - 3*$N/2*exp(-2*$X/$N)
            + 3*$N  *exp(  -$X/$N)
            + $X;
}

sub ex_4 {
   my ($N, $X) = @_;
    return     -$N/4*exp(-4*$X/$N)
            + 4*$N/3*exp(-3*$X/$N)
            - 3*$N  *exp(-2*$X/$N)
            + 4*$N  *exp(  -$X/$N)
            + $X;
}

sub ex_10 {
   my ($N, $X) = @_;
    return       -$N/10*exp(-10*$X/$N)
            +  10*$N/9 *exp( -9*$X/$N)
            -  45*$N/8 *exp( -8*$X/$N)
            + 120*$N/7 *exp( -7*$X/$N)
            -  35*$N   *exp( -6*$X/$N)
            + 252*$N/5 *exp( -5*$X/$N)
            - 105*$N/2 *exp( -4*$X/$N)
            +  40*$N   *exp( -3*$X/$N)
            -  45*$N/2 *exp( -2*$X/$N)
            +  10*$N   *exp( -1*$X/$N)
            + $X;
}