Editing Geometric distribution (section)

==Related distributions==

* The sum of <math>r</math> [[Statistical Independence|independent]] geometric random variables with parameter <math>p</math> is a [[Negative binomial distribution|negative binomial]] random variable with parameters <math>r</math> and <math>p</math>.<ref>{{Cite book |last=Pitman |first=Jim |url=http://link.springer.com/10.1007/978-1-4612-4374-8 |title=Probability |date=1993 |publisher=Springer New York |isbn=978-0-387-94594-1 |location=New York, NY |page=372 |language=en |doi=10.1007/978-1-4612-4374-8}}</ref> The geometric distribution is a special case of the negative binomial distribution, with <math>r=1</math>.

*The geometric distribution is a special case of discrete [[compound Poisson distribution]].<ref name=":9" />{{Rp|page=606}}
* The minimum of <math>n</math> geometric random variables with parameters <math>p_1, \dotsc, p_n</math> is also geometrically distributed with parameter <math>1 - \prod_{i=1}^n (1-p_i)</math>.<ref>{{cite journal |last1=Ciardo |first1=Gianfranco |last2=Leemis |first2=Lawrence M. |last3=Nicol |first3=David |date=1 June 1995 |title=On the minimum of independent geometrically distributed random variables |url=https://dx.doi.org/10.1016/0167-7152%2894%2900130-Z |journal=Statistics & Probability Letters |language=en |volume=23 |issue=4 |pages=313–326 |doi=10.1016/0167-7152(94)00130-Z |s2cid=1505801 |hdl-access=free |hdl=2060/19940028569}}</ref>

* Suppose 0&nbsp;<&nbsp;''r''&nbsp;<&nbsp;1, and for ''k''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;... the random variable ''X''<sub>''k''</sub> has a [[Poisson distribution]] with expected value ''r''<sup>''k''</sup>/''k''.  Then

::<math>\sum_{k=1}^\infty k\,X_k</math>

:has a geometric distribution taking values in <math>\mathbb{N}_0</math>, with expected value ''r''/(1&nbsp;&minus;&nbsp;''r'').{{citation needed|date=May 2012}}

* The [[exponential distribution]] is the continuous analogue of the geometric distribution. Applying the [[Floor and ceiling functions|floor]] function to the exponential distribution with parameter <math>\lambda</math> creates a geometric distribution with parameter <math>p=1-e^{-\lambda}</math> defined over <math>\mathbb{N}_0</math>.<ref name=":2" />{{Rp|page=74}} This can be used to generate geometrically distributed random numbers as detailed in [[Geometric distribution#Random variate generation|§ Random variate generation]].

* If ''p'' = 1/''n'' and ''X'' is geometrically distributed with parameter ''p'', then the distribution of ''X''/''n'' approaches an [[exponential distribution]] with expected value 1 as ''n''&nbsp;&rarr;&nbsp;&infin;, since<math display="block">
\begin{align}
\Pr(X/n>a)=\Pr(X>na) & = (1-p)^{na} = \left(1-\frac 1 n \right)^{na} = \left[ \left( 1-\frac 1 n \right)^n \right]^{a} \\
& \to [e^{-1}]^{a} = e^{-a} \text{ as } n\to\infty.
\end{align}
</math>More generally, if ''p''&nbsp;=&nbsp;''λ''/''n'', where ''λ'' is a parameter, then as ''n''&rarr;&nbsp;&infin; the distribution of ''X''/''n'' approaches an exponential distribution with rate ''λ'':<math>\Pr(X>nx)=\lim_{n \to \infty}(1-\lambda /n)^{nx}=e^{-\lambda x}</math> therefore the distribution function of ''X''/''n'' converges to <math>1-e^{-\lambda x}</math>, which is that of an exponential random variable.{{Cn|date=July 2024}}
* The [[index of dispersion]] of the geometric distribution is <math>\frac{1}{p}</math> and its [[coefficient of variation]] is <math>\frac{1}{\sqrt{1-p}}</math>. The distribution is [[Overdispersion|overdispersed]].<ref name=":8" />{{Rp|page=216}}