[OT] Generating distribution of N dice rolls

[Siarhei Siamashka]

On Thursday, 10 November 2022 at 23:15:24 UTC, H. S. Teoh wrote:
> According to the Wikipedia page on multinomial distribution 
> (linked by
> Timon), it states that the variance of X_i for n rolls of a 
> k-sided dice
> (with probability p_i), where i is a specific outcome, is:
>
> 	Var(X_i) = n*p_i*(1 - p_i)
>
> Don't really understand where this formula came from (as I 
> said, that page is way above my head), but we can make use of 
> it.

This is where things take a wrong turn. In reality you need more 
than just a matching mean and variance to correctly simulate some 
arbitrary probability distribution: 
https://en.wikipedia.org/wiki/Moment_(mathematics)

Every n-th moment needs to be correct too. Some of these moments 
have special names (n=1 mean, n=2 variance, n=3 skewness, n=4 
kurtosis, ...). If you only take care of the mean and variance 
for simulating a random distribution, then it's somewhat similar 
to approximating "sin(x) = x - (x^3 / 3!)" via taking only the 
first few terms of the Taylor series.

I wonder what's the reason for not using the mir-random library 
like suggested in the early comments? Do you want to avoid having 
an extra dependency?

```D
/+dub.sdl:
dependency "mir-random" version="~>2.2.19"
+/
import std, mir.random.engine, mir.random.ndvariable;

uint[k] diceDistrib(uint k)(uint N)
   in(k > 0)
   in(N > 0)
   out(r; r[].sum == N)
{
   uint[k] result;
   double[k] p;
   p[] = 1.0 / k;
   auto rv = multinomialVar(N, p);
   rv(rne, result[]);
   return result;
}

void main()
{
   writeln(diceDistrib!6(100_000_000));
}
```