Editing Negative binomial distribution (section)

===Representation as compound Poisson distribution===
The negative binomial distribution {{math|NB(''r'', ''p'')}} can be represented as a [[compound Poisson distribution]]: Let <math display=inline> (Y_n)_{n\,\in\,\mathbb N} </math> denote a sequence of [[independent and identically distributed random variables]], each one having the [[logarithmic distribution|logarithmic series distribution]] {{math|Log(''p'')}}, with probability mass function

: <math> f(k; r, p) =  \frac{-p^k}{k\ln(1-p)},\qquad k\in{\mathbb N}.</math>

Let {{mvar|N}} be a random variable, [[independence (probability theory)|independent]] of the sequence, and suppose that {{mvar|N}} has a [[Poisson distribution]] with mean {{math|λ {{=}} −''r'' ln(1 − ''p'')}}. Then the random sum

: <math>X=\sum_{n=1}^N Y_n</math>

is {{math|NB(''r'', ''p'')}}-distributed. To prove this, we calculate the [[probability generating function]] {{math|''G''{{sub|''X''}}}} of {{mvar|X}}, which is the composition of the probability generating functions {{math|''G''{{sub|''N''}}}} and {{math|''G''{{sub|''Y''{{sub|1}}}}}}. Using

:<math>G_N(z)=\exp(\lambda(z-1)),\qquad z\in\mathbb{R},</math>

and

: <math>G_{Y_1}(z)=\frac{\ln(1-pz)}{\ln(1-p)},\qquad |z|<\frac1p,</math>

we obtain

: <math>
\begin{align}G_X(z) & =G_N(G_{Y_1}(z))\\[4pt]
&=\exp\biggl(\lambda\biggl(\frac{\ln(1-pz)}{\ln(1-p)}-1\biggr)\biggr)\\[4pt]
&=\exp\bigl(-r(\ln(1-pz)-\ln(1-p))\bigr)\\[4pt]
&=\biggl(\frac{1-p}{1-pz}\biggr)^r,\qquad |z|<\frac1p,
\end{align}
</math>

which is the probability generating function of the {{math|NB(''r'', ''p'')}} distribution.

The following table describes four distributions related to the number of successes in a sequence of draws:
{| class="wikitable"
|-
!  !! With replacements !! No replacements
|-
| Given number of draws || [[binomial distribution]] || [[hypergeometric distribution]]
|-
| Given number of failures || negative binomial distribution || [[negative hypergeometric distribution]]
|}