Editing Binomial distribution (section)

=== Poisson approximation ===

The binomial distribution converges towards the [[Poisson distribution]] as the number of trials goes to infinity while the product {{math|''np''}} converges to a finite limit. Therefore, the Poisson distribution with parameter {{math|1=''λ'' = ''np''}} can be used as an approximation to {{math|B(''n'', ''p'')}} of the binomial distribution if {{math|''n''}} is sufficiently large and {{math|''p''}} is sufficiently small.  According to rules of thumb, this approximation is good if {{math|''n'' ≥ 20}} and {{math|''p'' ≤ 0.05}}<ref>{{cite news |date=2023-03-28 |title=12.4 – Approximating the Binomial Distribution {{!}} STAT 414 |newspaper=Pennstate: Statistics Online Courses |url=https://online.stat.psu.edu/stat414/lesson/12/12.4 |access-date=2023-10-08 |archive-date=2023-03-28 |archive-url=https://web.archive.org/web/20230328081322/https://online.stat.psu.edu/stat414/lesson/12/12.4 |url-status=bot: unknown }}</ref> such that {{math|''np'' ≤ 1}}, or if {{math|''n'' > 50}} and {{math|''p'' < 0.1}} such that {{math|''np'' < 5}},<ref>{{Cite book |last=Chen |first=Zac |title=H2 mathematics handbook |publisher=Educational publishing house |year=2011 |isbn=9789814288484 |edition=1 |location=Singapore |pages=348}}</ref> or if {{math|''n'' ≥ 100}} and {{math|''np'' ≤ 10}}.<ref name="nist">[[NIST]]/[[SEMATECH]], [http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc331.htm "6.3.3.1. Counts Control Charts"], ''e-Handbook of Statistical Methods.''</ref><ref>{{Cite web |date=2023-03-13 |title=The Connection Between the Poisson and Binomial Distributions |url=https://mathcenter.oxford.emory.edu/site/math117/connectingPoissonAndBinomial/ |access-date=2023-10-08 |archive-date=2023-03-13 |archive-url=https://web.archive.org/web/20230313085931/https://mathcenter.oxford.emory.edu/site/math117/connectingPoissonAndBinomial/ |url-status=bot: unknown }}</ref>

Concerning the accuracy of Poisson approximation, see Novak,<ref>Novak S.Y. (2011) Extreme value methods with applications to finance. London: CRC/ Chapman & Hall/Taylor & Francis. {{ISBN|9781-43983-5746}}.</ref> ch. 4, and references therein.