How to Generate Random Numbers with a Fixed Sum, Guaranteed Uniform Distribution?

Generating Random Numbers with a Fixed Sum

The challenge posed is to generate a series of pseudo-random numbers whose sum equals a predefined value. Specifically, how to generate four numbers that, when added together, equal 40.

Instead of relying on a method that could bias the distribution of the first number, a more uniform approach is employed. The solution utilizes a strategy of dividing the predefined value into smaller segments, using randomly selected dividers.

Assume we have four random positive integers (e, f, g, and h) such that 0 < e < f < g < h < 40. We can derive the desired four numbers as:

a = e
b = f - e
c = g - f
d = 40 - g

This technique guarantees an equal probability for each set of numbers, ensuring a uniform distribution. The resulting random numbers meet the requirement of summing to the predefined value.

Extending this concept, the following Python function generates a random list of positive integers summing to a specified total:

<code class="python">import random

def constrained_sum_sample_pos(n, total):
    """Return a randomly chosen list of n positive integers summing to total.
    Each such list is equally likely to occur."""

    dividers = sorted(random.sample(range(1, total), n - 1))
    return [a - b for a, b in zip(dividers + [total], [0] + dividers)]</code>

To generate non-negative integers, an additional transformation is employed:

<code class="python">def constrained_sum_sample_nonneg(n, total):
    """Return a randomly chosen list of n nonnegative integers summing to total.
    Each such list is equally likely to occur."""

    return [x - 1 for x in constrained_sum_sample_pos(n, total + n)]</code>

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