Adjust function for pyramid-like distribution - php

Question

In this question I got help to write a PHP function which gives a pyramid-like distribution:
function getRandomStrength($min, $max) {
$ln_low = log($min, M_E);
$ln_high = log($max, M_E);
$scale = $ln_high-$ln_low;
$rand = (mt_rand()/mt_getrandmax())*$scale+$ln_low;
$value = round(pow(M_E, $rand), 1);
return $value;
}
getRandomStrenth(1.1, 9.9);
// output could be: 1.4 or 8.3 or 9.8 or 7.2 or 2.9 or ...
When I run 50,000 iterations and check how often the numbers from 1 to 9 appear, I get the following list:
1 » 26%
2 » 19%
3 » 14%
4 » 10%
5 » 9%
6 » 7%
7 » 6%
8 » 6%
9 » 4%
This is what I wanted to have. But now I would like to adjust this function a bit. The smaller values should appear more often and the big values should appear less often - so that I get a list like this:
1 » 28%
2 » 20%
3 » 15%
4 » 11%
5 » 9%
6 » 6%
7 » 5%
8 » 5%
9 » 2%
As you can see, I just need a slight modification. But what can I change so that my function behaves as expected?
I tried several things (e.g. changing the base of the logarithm) but this did not change anything.

Answer 1

You can use pow on the random number.
$rand = pow( mt_rand()/mt_getrandmax(), 1.2 )*$scale+$ln_low;
By playing with the exponent value, you can get less or more small value.

Answer 2

Reducing the $scale of your function by a small (constant) amount seems to generate results pretty close to what you're looking for. You can achieve more accurate results by making this reduction of $scale a function of the randomly generated number from mt_rand(), which would require saving (mt_rand()/mt_getrandmax()) to a variable and performing some additional math on $scale.
Here are my tests, you can run it yourself: http://codepad.viper-7.com/ssblbQ
function getRandomStrength($min, $max)
{
$ln_low = log($min, M_E);
$ln_high = log($max, M_E);
$scale = $ln_high-$ln_low - .05; // Subtract a small constant, vary between .05 and .08
$rand = (mt_rand()/mt_getrandmax())*$scale+$ln_low;
$value = round(pow(M_E, $rand), 1);
return $value;
}
$values = array_fill(1, 9, 0);
for( $i = 0; $i < 50000; $i++)
{
$values[ intval( getRandomStrength(1.1, 9.9)) ]++;
}
for( $i = 1; $i <= 9; $i++)
{
$values[ $i] /= 500; // / 50000 * 100 to get a percent
}
var_dump( $values);
Output
Run #1 - Constant = 0.5
array(9) {
[1] => float(26.626) // Should be 28
[2] => float(19.464) // Should be 20
[3] => float(13.476) // Should be 15
[4] => float(10.41) // Should be 11
[5] => float(8.616) // Should be 9
[6] => float(7.198) // Should be 6
[7] => float(6.258) // Should be 5
[8] => float(5.52) // Should be 5
[9] => float(2.432) // Should be 2
}
Run #2 - Constant = 0.65
array(9) {
[1] => float(26.75) // Should be 28
[2] => float(19.466) // Should be 20
[3] => float(13.872) // Should be 15
[4] => float(10.562) // Should be 11
[5] => float(8.466) // Should be 9
[6] => float(7.222) // Should be 6
[7] => float(6.454) // Should be 5
[8] => float(5.554) // Should be 5
[9] => float(1.654) // Should be 2
}
Run #3 - Constant = 0.70
array(9) {
[1] => float(26.848) // Should be 28
[2] => float(19.476) // Should be 20
[3] => float(13.808) // Should be 15
[4] => float(10.764) // Should be 11
[5] => float(8.67) // Should be 9
[6] => float(7.148) // Should be 6
[7] => float(6.264) // Should be 5
[8] => float(5.576) // Should be 5
[9] => float(1.446) // Should be 2
}

Answer 3

For n in {0..1}, y=(x^n)-1, y will range from 0 to x-1. That curve is then easily mapped from 0 to some max value by multiplying by the range and dividing by (x-1). If you change the value x to something near one, the curve will be nearly linear, and at large values, the curve becomes more like a hockey-stick, but will still fall in the same range.
My initial sample value of three won't be precisely what you expressed, but you can adjust it to get the distribution curve you're looking for.
function getCustomStrength($min, $max, $x_val, $base) {
$logmax = $base-1;
$range = $max-$min;
return (pow($base,$x_val)-1)*($range/($base-1))+$min;
}
function getRandomStrength($min, $max) {
$rand = mt_rand()/mt_getrandmax();
$base = 3.0;
return getCustomStrength($min, $max, $rand, $base);
}
getRandomStrength(1.1, 9.9);