Template:Negative binomial distribution: Difference between revisions

Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure, so identifying the specific parametrization used is crucial in any given text.
	Probability mass functionFile:Negbinomial.gif; The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation.
Notation	NB(r,p)
Parameters	r > 0 — number of successes until the experiment is stopped (integer, but the definition can also be extended to reals); p ∈ [0,1] — success probability in each experiment (real)
Support	k ∈ { 0, 1, 2, 3, … } — number of failures
PMF	k↦(k+r−1k)⋅(1−p)kpr, involving a binomial coefficient
CDF	k↦Ip(r,k+1), the regularized incomplete beta function
Mean	r(1−p)p
Mode	{⌊(r−1)(1−p)p⌋if r>10if r≤1
Variance	r(1−p)p2
Skewness	2−p(1−p)r
Excess kurtosis	6r+p2(1−p)r
MGF	(p1−(1−p)et)r for t<−log⁡(1−p)
CF	(p1−(1−p)eit)r with t∈ℝ
PGF	(p1−(1−p)z)r for \|z\|<1p
Fisher information	rp2(1−p)
Method of moments	r=E[X]2V[X]−E[X] ; p=E[X]V[X]

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Latest revision as of 01:11, 23 February 2025

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Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure,^[1] so identifying the specific parametrization used is crucial in any given text.
Probability mass function File:Negbinomial.gif The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation.
Notation	$N B (r, p)$
Parameters	r > 0 — number of successes until the experiment is stopped (integer, but the definition can also be extended to reals) p ∈ [0,1] — success probability in each experiment (real)
Support	k ∈ { 0, 1, 2, 3, … } — number of failures
PMF	$k \mapsto (\binom{k + r - 1}{k}) \cdot (1 - p)^{k} p^{r},$ involving a binomial coefficient
CDF	$k \mapsto I_{p} (r, k + 1),$ the regularized incomplete beta function
Mean	$\frac{r (1 - p)}{p}$
Mode	${\begin{cases} ⌊ \frac{(r - 1) (1 - p)}{p} ⌋ & if r > 1 \\ 0 & if r \leq 1 \end{cases}$
Variance	$\frac{r (1 - p)}{p^{2}}$
Skewness	$\frac{2 - p}{\sqrt{(1 - p) r}}$
Excess kurtosis	$\frac{6}{r} + \frac{p^{2}}{(1 - p) r}$
MGF	$(\frac{p}{1 - (1 - p) e^{t}})^{r} for t < - \log (1 - p)$
CF	$(\frac{p}{1 - (1 - p) e^{i t}})^{r} with t \in ℝ$
PGF	$(\frac{p}{1 - (1 - p) z})^{r} for \| z \| < \frac{1}{p}$
Fisher information	$\frac{r}{p^{2} (1 - p)}$
Method of moments	$r = \frac{E [X]^{2}}{V [X] - E [X]}$ $p = \frac{E [X]}{V [X]}$