Negative Binomial equations

· Format:

Negbinomial(p, s)

· Probability mass function:

$\begin{array}{l}f(x)=\left( \begin{array}{c} x-1 \\ x-s \end{array} \right) p^s(1-p)^{x-s}\end{array}$

· Cumulative distribution function:

$\begin{array}{l}F(x)=\displaystyle\sum_{i=s}^{\lfloor x \rfloor} \left( \begin{array}{c} x-1 \\ i-s \end{array} \right) p^s(1-p)^{i-s}\end{array}$

· Parameter restrictions:

s > 0, s is an integer, 0 < p ≤ 1

· Domain:

x = {s, s+1, s+2,...}

· Mean:

$\begin{array}{l}\frac{s}{p}\end{array}$

· Mode:

z, z +1

if z is an integer

|z|+ 1

otherwise

where z = (s - 1)/ p

· Variance:

$\begin{array}{l}\displaystyle \frac{s(1-p)}{p^2}\end{array}$

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