Editing Geometric distribution (section)

=== Bayesian inference ===
In [[Bayesian inference]], the parameter <math>p</math> is a random variable from a [[prior distribution]] with a [[posterior distribution]] calculated using [[Bayes' theorem]] after observing samples.<ref name=":3" />{{Rp|page=167}} If a [[beta distribution]] is chosen as the prior distribution, then the posterior will also be a beta distribution and it is called the [[conjugate distribution]]. In particular, if a <math>\mathrm{Beta}(\alpha,\beta)</math> prior is selected, then the posterior, after observing samples <math>k_1, \dotsc, k_n \in \mathbb{N}</math>, is<ref>{{Cite CiteSeerX |citeseerx=10.1.1.157.5540 |first=Daniel |last=Fink |title=A Compendium of Conjugate Priors}}</ref><math display="block">p \sim \mathrm{Beta}\left(\alpha+n,\ \beta+\sum_{i=1}^n (k_i-1)\right). \!</math>Alternatively, if the samples are in <math>\mathbb{N}_0</math>, the posterior distribution is<ref>{{Cite web|url=http://halweb.uc3m.es/esp/Personal/personas/mwiper/docencia/English/PhD_Bayesian_Statistics/ch3_2009.pdf |archive-url=https://web.archive.org/web/20100408092905/http://halweb.uc3m.es/esp/Personal/personas/mwiper/docencia/English/PhD_Bayesian_Statistics/ch3_2009.pdf |archive-date=2010-04-08 |url-status=live|title=3. Conjugate families of distributions}}</ref><math display="block">p \sim \mathrm{Beta}\left(\alpha+n,\beta+\sum_{i=1}^n k_i\right).</math>Since the expected value of a <math>\mathrm{Beta}(\alpha,\beta)</math> distribution is <math>\frac{\alpha}{\alpha+\beta}</math>,<ref name=":9" />{{Rp|page=145}} as <math>\alpha</math> and <math>\beta</math> approach zero, the posterior mean approaches its maximum likelihood estimate.