Function malachite_base::num::random::geometric::geometric_random_negative_signeds

pub fn geometric_random_negative_signeds<T: PrimitiveSigned>(
    seed: Seed, 
    abs_um_numerator: u64, 
    abs_um_denominator: u64
) -> GeometricRandomNegativeSigneds<T>

Generates random negative signed integers from a modified geometric distribution.

This distribution can be derived from a truncated geometric distribution by negating its domain. The distribution is truncated at T::MIN.

With this distribution, the probability of a value being generated decreases as its absolute value increases. The probabilities $P(-1), P(-2), P(-3), \ldots$ decrease in a geometric sequence; that’s where the “geometric” comes from. Values below T::MIN 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 abs_um_numerator / abs_um_denominator. The unadjusted mean is what the mean of the absolute values of the generated values would be if the distribution were not truncated. If $m_u$ is significantly lower than -T::MIN, which is usually the case, then it is very close to the actual mean of the absolute values. 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 one. It may be arbitrarily high, but note that the iteration time increases linearly with abs_um_numerator + abs_um_denominator.

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

and whenever $n, n - 1 \in S$, $$ \frac{P(n)}{P(n-1)} = \frac{m_u}{m_u - 1}. $$

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 $n$ = abs_um_numerator + abs_um_denominator.

Panics

Panics if abs_um_denominator is zero or if abs_um_numerator <= abs_um_denominator.

Examples

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

assert_eq!(
    prefix_to_string(geometric_random_negative_signeds::<i64>(EXAMPLE_SEED, 2, 1), 10),
    "[-2, -1, -1, -4, -5, -5, -2, -1, -1, -2, ...]"
)

Further details

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

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

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