gleam_stats - v1.1.0

gleam_stats/distributions/binomial

Functions related to discrete binomial random variables.

Functions

binomial_cdf

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pub fn binomial_cdf(x: Int, n: Int, p: Float) -> Result(
  Float,
  String,
)

Evaluate, at a certain point $$x \in \mathbb{Z}$$, the cumulative distribution function (cdf) of a discrete binomial random variable with parameters $$n \in \mathbb{Z}_{\geq 0}$$ (number of trials) and $$p \in [0, 1]$$ (the success probability in each trial).

The cdf is defined as:

\[ F(x; n, p) = \begin{cases} \sum_{i=0}^{x} \binom{n}{i} \cdot p^{i} \cdot (1-p)^{n-i} &\text{if } x \geq 0,\\ 0 &\text{if } x < 0. \end{cases} \]

Example: 
 import gleam_stats/distributions/binomial
 import gleeunit/should

 pub fn example() {
   let n: Float = 40.
   let p: Float = 0.5
   // For illustrational purposes, evaluate the cdf at the 
   // point -100.0
   binomial.binomial_cdf(-100.0, n, p) 
   |> should.equal(Ok(0.0))
 }

binomial_mean

</> 
pub fn binomial_mean(n: Int, p: Float) -> Result(Float, String)

Analytically compute the mean of a discrete binomial random variable with parameters $$n \in \mathbb{Z}_{\geq 0}$$ (number of trials) and $$p \in [0, 1]$$ (the success probability in each trial).

The mean returned is: $$n \cdot p$$.

binomial_pmf

</> 
pub fn binomial_pmf(x: Int, n: Int, p: Float) -> Result(
  Float,
  String,
)

Evaluate, at a certain point $$x \in \mathbb{Z}$$, the probability mass function (pmf) of a discrete binomial random variable with parameters $$n \in \mathbb{Z}_{\geq 0}$$ (number of trials) and $$p \in [0, 1]$$ (the success probability in each trial).

The pmf is defined as:

\[ f(x; n, p) = \begin{cases} \binom{n}{x} \cdot p^{x} \cdot (1 - p)^{n - x} &\text{if } x \geq 0, \\ 0 &\text{if } x < 0. \end{cases} \]

Example: 
 import gleam_stats/distributions/binomial
 import gleeunit/should

 pub fn example() {
   let n: Float = 40.
   let p: Float = 0.5
   // For illustrational purposes, evaluate the pmf at the 
   // point -100.0
   binomial.binomial_pmf(-100.0, n, p) 
   |> should.equal(Ok(0.0))
 }

binomial_random

</> 
pub fn binomial_random(stream: Iterator(Int), n: Int, p: Float, m: Int) -> Result(
  #(List(Int), Iterator(Int)),
  String,
)

Generate $$m \in \mathbb{Z}_{>0}$$ random numbers from a binomial distribution (discrete) with parameters $$n \in \mathbb{Z}_{\geq 0}$$ (number of trials) and $$p \in [0, 1]$$ (the success probability in each trial).

The random numbers are generated using the inverse transform method.

Example: 
 import gleam/iterator.{Iterator}
 import gleam_stats/generators
 import gleam_stats/distributions/binomial

 pub fn example() {
   let seed: Int = 5
   let seq: Int = 1
   let n: Float = 40.
   let p: Float = 0.5
   assert Ok(out) =
     generators.seed_pcg32(seed, seq)
     |> binomial.binomial_random(n, p, 5_000)
   let rands: List(Float) = pair.first(out)
   let stream: Iterator(Int) = pair.second(out)
 }

binomial_variance

</> 
pub fn binomial_variance(n: Int, p: Float) -> Result(
  Float,
  String,
)

Analytically compute the variance of a discrete binomial random variable with parameters $$n \in \mathbb{Z}_{\geq 0}$$ (number of trials) and $$p \in [0, 1]$$ (the success probability in each trial).

The variance returned is: $$n \cdot p \cdot (1 - p)$$.

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