:Template:Negative binomial distribution

{{Probability distribution

| intro = 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,{{cite book|last = DeGroot| first = Morris H.| author-link = Morris H. DeGroot| title = Probability and Statistics| edition = Second| year = 1986| publisher = Addison-Wesley| isbn = 0-201-11366-X| oclc = 10605205| lccn = 84006269 |pages = 258–259}} so identifying the specific parametrization used is crucial in any given text.

| type = mass

| pdf_image = 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 = $\mathrm{NB}(r,\,p)$

| 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

| pdf = $k\mapsto{k+r-1 \choose k}\cdot (1-p)^k p^r,$ involving a binomial coefficient

| cdf = $k\mapsto I_p(r,\,k+1),$ the regularized incomplete beta function

| mean = $\frac{r(1-p)}{p}$

| median =

| mode = $\begin{cases}\left\lfloor\frac{(r-1)(1-p)}{p}\right\rfloor & \text{if } r>1 \\
0 & \text{if } r\leq 1\end{cases}$

| variance = $\frac{r(1-p)}{p^2}$

| skewness = $\frac{2-p}{\sqrt{(1-p)r}}$

| kurtosis = $\frac{6}{r} + \frac{p^2}{(1-p)r}$

| entropy =

| mgf = $\biggl(\frac{p}{1 - (1 - p)e^t}\biggr)^{\!r} \text{ for }t<-\log (1-p)$

| char = $\biggl(\frac{p}{1 - (1 - p)e^{i\,t}}\biggr)^{\!r} \text{ with }t\in\mathbb{R}$

| pgf = $\biggl(\frac{p}{1 - (1 - p)z}\biggr)^{\!r} \text{ for }|z|<\frac1p$

| fisher = $\frac{r}{p^2 (1-p) }$

| moments = $r = \frac{E[X]^2}{V[X] - E[X]}$
$p = \frac{E[X]}{V[X]}$

}}

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