How to generate a random number, the larger the number, the less chance?

Question

PHP

It is necessary to generate a random number in a certain range, while the chance for a larger number should be less (The larger the number, the less the chance of it falling out)

Answer 1

Uh, a problem from school mathematics, I would like to remember more.
If I think correctly, we need to take the probability density function, integrate over a given interval, take the inverse function from the integral and substitute the arguments with a uniform distribution generator for this function.
Well, that is, if you have a linear decrease in the probability f (x) = -x, it will be a rotated square root on a given interval (something like, count who has not forgotten the integrals from school yet).
In general it turns out:

/**
 * Generates a random number between 0.00 and 1.00
 */
function generateNum()
{
    return 1 - sqrt(rand(0, 1000)/1000);
}

Test:

$nums = [];
for($i = 0; $i < 10000; ++$i) {
    $index = sprintf('%.1f', floor(generateNum() * 10) / 10);
    $nums[$index] = ($nums[$index] ?? 0) + 1;
}

ksort($nums);

var_dump($nums);

The resulting probability density (frequency):

Answer 2

If you multiply two random numbers, then you will no longer get a linear distribution, but a Pearson distribution.

You can play around with the numbers by adding some coefficients.

Answer 3

just loop from 1 to your max, generate either 0 or 1 randomly. If the roll is 1, then continue the loop. If 0 is dropped, then stop the loop. The result will be the current iteration of the loop. Thus, the probability of each larger number decreases.