NEGATIVE BINOMIAL DISTRIBUTION

Phitter implementation

Distribution Definition

python

import phitter

distribution = phitter.discrete.NegativeBinomial({"r": *, "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 NegativeBinomial (r, p)

Distribution Domain

x \in N \equiv {0, 1, 2, \dots}

Parameters Domain and Constraints

r \in N_{⩾ 1}, p \in (0, 1) \subseteq R

Cumulative Distribution Function

F_{X} (x) = I (p, r, x + 1)

Probability Density Function

f_{X} (x) = (\binom{r + x - 1}{x}) p^{r} (1 - p)^{x}

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 = 0}^{\infty} x^{k} f_{X} (x)

Parametric Mean

Mean (X) = μ_{1}^{'} = \frac{r (1 - p)}{p}

Parametric Variance

Variance (X) = (μ_{2}^{'} - μ_{1}^{' 2}) = \frac{r (1 - p)}{p^{2}}

Parametric Skewness

Skewness (X) = \frac{μ_{3}^{'} - 3 μ_{2}^{'} μ_{1}^{'} + 2 μ_{1}^{' 3}}{(μ_{2}^{'} - μ_{1}^{' 2})^{1.5}} = \frac{2 - p}{\sqrt{r (1 - p)}}

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{6}{r} + \frac{p^{2}}{r (1 - p)}

Parametric Median

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

Parametric Mode

Mode (X) = ⌊ (r - 1) (1 - p) / p ⌋

Additional Information and Definitions

$Computing an analytic expression for the inverse of the cumulative distribution function$ $is not feasible. However, it is possible to calculate the Percentile Point Function by$ $approximating it to the nearest integer.$
$u : Uniform[0,1] random varible$
$I (x, a, b) : Regularized incomplete beta function$
$⌊ x ⌋ : Floor function$

Spreadsheet Documents

NEGATIVE BINOMIAL DISTRIBUTION ​

Phitter implementation ​

Equations ​

NEGATIVE BINOMIAL DISTRIBUTION

Phitter implementation

Equations