Let the random variable follow a negative binomial distribution with mean 5. Then its variance could be
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Negative binomial distribution
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In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs. For example, if we define a 1 as failure, all non-1s as successes, and we throw a dice repeatedly until 1 appears the third time (r = three failures), then the probability distribution of the number of non-1s that appeared will be a negative binomial distribution.
Different texts 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 it is crucial to identify the specific parametrization used in any given text.
Probability mass function
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
{\displaystyle \mathrm {NB} (r,\,p)}\mathrm {NB} (r,\,p)
Parameters
r > 0 — number of failures 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 successes
pmf
{\displaystyle k\mapsto {k+r-1 \choose k}\cdot (1-p)^{r}p^{k},}{\displaystyle k\mapsto {k+r-1 \choose k}\cdot (1-p)^{r}p^{k},} involving a binomial coefficient
CDF
{\displaystyle k\mapsto 1-I_{p}(k+1,\,r),}{\displaystyle k\mapsto 1-I_{p}(k+1,\,r),} the regularized incomplete beta function
Mean
{\displaystyle {\frac {pr}{1-p}}}{\frac {pr}{1-p}}
Mode
{\displaystyle {\begin{cases}{\big \lfloor }{\frac {p(r-1)}{1-p}}{\big \rfloor }&{\text{if}}\ r>1\\0&{\text{if}}\ r\leq 1\end{cases}}}{\begin{cases}{\big \lfloor }{\frac {p(r-1)}{1-p}}{\big \rfloor }&{\text{if}}\ r>1\\0&{\text{if}}\ r\leq 1\end{cases}}
Variance
{\displaystyle {\frac {pr}{(1-p)^{2}}}}{\frac {pr}{(1-p)^{2}}}
Skewness
{\displaystyle {\frac {1+p}{\sqrt {pr}}}}{\frac {1+p}{\sqrt {pr}}}
Ex. kurtosis
{\displaystyle {\frac {6}{r}}+{\frac {(1-p)^{2}}{pr}}}{\frac {6}{r}}+{\frac {(1-p)^{2}}{pr}}
MGF
{\displaystyle {\biggl (}{\frac {1-p}{1-pe^{t}}}{\biggr )}^{\!r}{\text{ for }}t<-\log p}{\biggl (}{\frac {1-p}{1-pe^{t}}}{\biggr )}^{\!r}{\text{ for }}t<-\log p
CF
{\displaystyle {\biggl (}{\frac {1-p}{1-pe^{i\,t}}}{\biggr )}^{\!r}{\text{ with }}t\in \mathbb {R} }{\biggl (}{\frac {1-p}{1-pe^{i\,t}}}{\biggr )}^{\!r}{\text{ with }}t\in \mathbb {R}
PGF
{\displaystyle {\biggl (}{\frac {1-p}{1-pz}}{\biggr )}^{\!r}{\text{ for }}|z|<{\frac {1}{p}}}{\biggl (}{\frac {1-p}{1-pz}}{\biggr )}^{\!r}{\text{ for }}|z|<{\frac {1}{p}}
Fisher information
{\displaystyle {\frac {r}{(1-p)^{2}p}}}{\displaystyle {\frac {r}{(1-p)^{2}p}}}
The Pascal distribution (after Blaise Pascal) and Polya distribution (for George Pólya) are special cases of the negative binomial distribution. A convention among engineers, climatologists, and others is to use "negative binomial" or "Pascal" for the case of an integer-valued stopping-time parameter r, and use "Polya" for the real-valued case.
For occurrences of "contagious" discrete events, like tornado outbreaks, the Polya distributions can be used to give more accurate models than the Poisson distribution by allowing the mean and variance to be different, unlike the Poisson. "Contagious" events have positively correlated occurrences causing a larger variance than if the occurrences were independent, due to a positive covariance term.
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