HYPERGEOMETRIC DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.discrete.Hypergeometric({"N": *, "K": *, "p": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

python

## CDF, PMF, PPF receive float or numpy.ndarray.
distribution.cdf(int | numpy.ndarray[int]) # -> float | numpy.ndarray
distribution.pmf(int | numpy.ndarray[int]) # -> float | numpy.ndarray
distribution.ppf(int | numpy.ndarray[int]) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> int
distribution.mode # -> int

Equations

Distribution Definition

X \sim Hypergeometric (N, K, n)

Distribution Domain

x \in {max (0, n + K - N), min (n, K)}

Parameters Domain and Constraints

N \in N, K \in {0 \dots N}, n \in {0 \dots N}

Cumulative Distribution Function

F_{X} (x) = \sum_{i = 0}^{x} (\binom{K}{i}) (\binom{N - K}{n - i}) / (\binom{N}{n})

Probability Density Function

f_{X} (x) = (\binom{K}{x}) (\binom{N - K}{n - x}) / (\binom{N}{n})

Percent Point Function / Sample

F_{X}^{- 1} (u) = \arg min_{x} | F_{X} (x) - u |

Parametric Centered Moments

E [X^{k}] = μ_{k}^{'} = \sum_{x = max (0, n + K - N)}^{min (n, K)} x^{k} f_{X} (x)

Parametric Mean

Mean (X) = μ_{1}^{'} = \frac{n K}{N}

Parametric Variance

Variance (X) = (μ_{2}^{'} - μ_{1}^{' 2}) = n \frac{K}{N} \frac{N - K}{N} \frac{N - n}{N - 1}

Parametric Skewness

Skewness (X) = \frac{μ_{3}^{'} - 3 μ_{2}^{'} μ_{1}^{'} + 2 μ_{1}^{' 3}}{(μ_{2}^{'} - μ_{1}^{' 2})^{1.5}} = \frac{(N - 2 K) (N - 1)^{\frac{1}{2}} (N - 2 n)}{[n K (N - K) (N - n)]^{\frac{1}{2}} (N - 2)}

Parametric Kurtosis

Kurtosis (X) = \frac{μ_{4}^{'} - 4 μ_{1}^{'} μ_{3}^{'} + 6 μ_{1}^{' 2} μ_{2}^{'} - 3 μ_{1}^{' 4}}{(μ_{2}^{'} - μ_{1}^{' 2})^{2}} = 3 + \frac{1}{n K (N - K) (N - n) (N - 2) (N - 3)}

Parametric Median

Median (X) = F_{X}^{- 1} (0.5)

Parametric Mode

Mode (X) = ⌊ \frac{(n + 1) (K + 1)}{N + 2} ⌋

Additional Information and Definitions

$Computing an analytic expression for the inverse of the cumulative distribution function$ $is not feasible. However, it is possible to calculate the Percentile Point Function by$ $approximating it to the nearest integer.$
$u : Uniform[0,1] random varible$
$⌊ x ⌋ : Floor function$
$⌈ x ⌉ : Ceiling Function$

Spreadsheet Documents

HYPERGEOMETRIC DISTRIBUTION ​

Phitter implementation ​

Equations ​

HYPERGEOMETRIC DISTRIBUTION

Phitter implementation

Equations