Editing Negative binomial distribution (section)

===Poisson distribution===
Consider a sequence of negative binomial random variables where the stopping parameter {{mvar|r}} goes to infinity, while the probability {{mvar|p}} of success in each trial goes to one, in such a way as to keep the mean of the distribution (i.e. the expected number of failures) constant. Denoting this mean as {{mvar|λ}}, the parameter {{mvar|p}} will be {{math|1=''p'' = ''r''/(''r'' + ''λ'')}}
: <math>
\begin{align}
    \text{Mean:} \quad & \lambda = \frac{(1-p)r}{p} \quad \Rightarrow \quad p = \frac{r}{r+\lambda}, \\
    \text{Variance:} \quad & \lambda \left( 1 + \frac{\lambda}{r} \right) > \lambda, \quad \text{thus always overdispersed}.
\end{align}
  </math>

Under this parametrization the probability mass function will be
: <math>
    f(k; r, p) = \frac{\Gamma(k+r)}{k!\cdot\Gamma(r)}(1-p)^k p^r = \frac{\lambda^k}{k!} \cdot \frac{\Gamma(r+k)}{\Gamma(r)\;(r+\lambda)^k} \cdot \frac{1}{\left(1+\frac{\lambda}{r}\right)^r}
  </math>

Now if we consider the limit as {{math|''r'' → ∞}}, the second factor will converge to one, and the third to the exponent function:
: <math>
    \lim_{r\to\infty} f(k; r, p) = \frac{\lambda^k}{k!} \cdot 1 \cdot \frac{1}{e^\lambda},
  </math>
which is the mass function of a [[Poisson distribution|Poisson-distributed]] random variable with expected value&nbsp;{{mvar|λ}}.

In other words, the alternatively parameterized negative binomial distribution [[convergence in distribution|converges]] to the Poisson distribution and {{mvar|r}} controls the deviation from the Poisson.  This makes the negative binomial distribution suitable as a robust alternative to the Poisson, which approaches the Poisson for large {{mvar|r}}, but which has larger variance than the Poisson for small {{mvar|r}}.
: <math>
    \operatorname{Poisson}(\lambda) = \lim_{r \to \infty} \operatorname{NB} \left(r, \frac{r}{r + \lambda}\right).
  </math>