Editing Negative binomial distribution (section)

===Distribution of a sum of geometrically distributed random variables===

If {{math|''Y''{{sub|''r''}}}} is a random variable following the negative binomial distribution with parameters {{mvar|r}} and {{mvar|p}}, and support {{math|{{mset|0, 1, 2, ...}}}}, then {{math|''Y''{{sub|''r''}}}} is a sum of {{mvar|r}} [[statistical independence|independent]] variables following the [[geometric distribution]] (on {{math|{{mset|0, 1, 2, ...}}}}) with parameter {{mvar|p}}. As a result of the [[central limit theorem]], {{math|''Y''{{sub|''r''}}}} (properly scaled and shifted) is therefore approximately [[normal distribution|normal]] for sufficiently large&nbsp;{{mvar|r}}.

Furthermore, if {{math|''B''{{sub|''s''+''r''}}}} is a random variable following the [[binomial distribution]] with parameters {{math|''s'' + ''r''}} and {{mvar|p}}, then

: <math>
\begin{align}
\Pr(Y_r \leq s) & {} = 1 - I_p(s+1, r) \\[5pt]
& {} = 1 - I_{p}((s+r)-(r-1), (r-1)+1) \\[5pt]
& {} = 1 - \Pr(B_{s+r} \leq r-1) \\[5pt]
& {} = \Pr(B_{s+r} \geq r) \\[5pt]
& {} = \Pr(\text{after } s+r \text{ trials, there are at least } r \text{ successes}).
\end{align}
</math>

In this sense, the negative binomial distribution is the "inverse" of the binomial distribution.

The sum of independent negative-binomially distributed random variables {{math|''r''{{sub|1}}}} and {{math|''r''{{sub|2}}}} with the same value for parameter {{mvar|p}} is negative-binomially distributed with the same {{mvar|p}} but with {{mvar|r}}-value&nbsp;{{math|''r''{{sub|1}} + ''r''{{sub|2}}}}.

The negative binomial distribution is [[Infinite divisibility (probability)|infinitely divisible]], i.e., if {{mvar|Y}} has a negative binomial distribution, then for any positive integer {{mvar|n}}, there exist independent identically distributed random variables {{math|''Y''{{sub|1}}, ..., ''Y''{{sub|''n''}}}} whose sum has the same distribution that {{mvar|Y}} has.