Function malachite_base::num::random::geometric::geometric_random_natural_signeds

pub fn geometric_random_natural_signeds<T: PrimitiveSigned>(
    seed: Seed,
    um_numerator: u64,
    um_denominator: u64
) -> GeometricRandomNaturalValues<T> 

Generates random natural (non-negative) signed integers from a truncated geometric distribution.

With this distribution, the probability of a value being generated decreases as the value increases. The probabilities $P(0), P(1), P(2), \ldots$ decrease in a geometric sequence; that’s where the “geometric” comes from. Unlike a true geometric distribution, this distribution is truncated, meaning that values above T::MAX are never generated.

The probabilities can drop more quickly or more slowly depending on a parameter $m_u$, called the unadjusted mean. It is equal to um_numerator / um_denominator. The unadjusted mean is what the mean generated value would be if the distribution were not truncated. If $m_u$ is significantly lower than T::MAX, which is usually the case, then it is very close to the actual mean. The higher $m_u$ is, the more gently the probabilities drop; the lower it is, the more quickly they drop. $m_u$ must be greater than zero. It may be arbitrarily high, but note that the iteration time increases linearly with um_numerator + um_denominator.

Here is a more precise characterization of this distribution. Let its support $S \subset \Z$ equal $[0, 2^{W-1})$, where $W$ is the width of the type. Then we have $$ P(n) \neq 0 \leftrightarrow n \in S $$ and whenever $n, n + 1 \in S$, $$ \frac{P(n)}{P(n+1)} = \frac{m_u + 1}{m_u}. $$

The output length is infinite.

Expected complexity per iteration

$T(n) = O(n)$

$M(n) = O(1)$

where $T$ is time, $M$ is additional memory, and $n$ = um_numerator + um_denominator.

Panics

Panics if um_numerator or um_denominator are zero, or, if after being reduced to lowest terms, their sum is greater than or equal to $2^{64}$.

Examples

use malachite_base::iterators::prefix_to_string;
use malachite_base::num::random::geometric::geometric_random_natural_signeds;
use malachite_base::random::EXAMPLE_SEED;

assert_eq!(
    prefix_to_string(geometric_random_natural_signeds::<i64>(EXAMPLE_SEED, 1, 1), 10),
    "[1, 0, 0, 3, 4, 4, 1, 0, 0, 1, ...]"
)

Further details

Geometric distributions are more typically parametrized by a parameter $p$. The relationship between $p$ and $m_u$ is $m_u = \frac{1}{p} - 1$, or $p = \frac{1}{m_u + 1}$.

The probability mass function of this distribution is $$ P(n) = \begin{cases} \frac{(1-p)^np}{1-(1-p)^{2^{W-1}}} & \text{if} \quad 0 \leq n < 2^{W-1}, \\ 0 & \text{otherwise}, \end{cases} $$ where $W$ is the width of the type.

It’s also useful to note that $$ \lim_{W \to \infty} P(n) = \begin{cases} (1-p)^np & \text{if} \quad n \geq 0, \\ 0 & \text{otherwise}. \end{cases} $$