Editing Negative binomial distribution (section)

===Probability mass function===

The [[probability mass function]] of the negative binomial distribution is
:<math>
    f(k; r, p) \equiv \Pr(X = k) = \binom{k+r-1}{k} (1-p)^k p^r
  </math>

where {{mvar|r}} is the number of successes, {{mvar|k}} is the number of failures, and {{mvar|p}} is the probability of success on each trial.

Here, the quantity in parentheses is the [[binomial coefficient]], and is equal to
:<math>
    \binom{k+r-1}{k} = \frac{(k+r-1)!}{(r-1)!\,(k)!} = \frac{(k+r-1)(k+r-2)\dotsm(r)}{k!} = \frac{\Gamma(k + r)}{k!\ \Gamma(r)}.
  </math>
Note that {{math|Γ(''r'')}} is the [[Gamma function]].

There are {{mvar|k}} failures chosen from {{math|''k'' + ''r'' − 1}} trials rather than {{math|''k'' + ''r''}} because the last of the {{math|''k'' + ''r''}} trials is by definition a success.

This quantity can alternatively be written in the following manner, explaining the name "negative binomial":

:<math>
\begin{align}
& \frac{(k+r-1)\dotsm(r)}{k!} \\[10pt]
= {} & (-1)^k \frac{\overbrace{(-r)(-r-1)(-r-2)\dotsm(-r-k+1)}^{k\text{ factors}}}{k!} = (-1)^k\binom{-r}{\phantom{-}k}.
\end{align}
</math>

Note that by the last expression and the [[binomial series]], for every {{math|0 ≤ ''p'' < 1}} and <math>q=1-p</math>,

:<math>
p^{-r} = (1-q)^{-r} = \sum_{k=0}^\infty \binom{-r}{\phantom{-}k}(-q)^k = \sum_{k=0}^\infty \binom{k+r-1}{k}q^k
</math>

hence the terms of the probability mass function indeed add up to one as below.
:<math>
\sum_{k=0}^\infty \binom{k+r-1}{k}(1-p)^kp^r = p^{-r}p^r = 1
</math>

To understand the above definition of the probability mass function, note that the probability for every specific sequence of {{mvar|r}}&nbsp;successes and {{mvar|k}}&nbsp;failures is {{math|''p''{{sup|''r''}}(1 − ''p''){{sup|''k''}}}}, because the outcomes of the {{math|''k'' + ''r''}} trials are supposed to happen [[independence (probability theory)|independently]]. Since the {{mvar|r}}-th success always comes last, it remains to choose the {{mvar|k}}&nbsp;trials with failures out of the remaining {{math|''k'' + ''r'' − 1}} trials. The above binomial coefficient, due to its combinatorial interpretation, gives precisely the number of all these sequences of length {{math|''k'' + ''r'' − 1}}.