Editing Binomial distribution (section)

=== Beta distribution ===

The binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials. The binomial distribution is the [[Probability mass function|PMF]] of {{mvar|k}} successes given {{mvar|n}} independent events each with a probability {{mvar|p}} of success. 
Mathematically, when {{math|1=''α'' = ''k'' + 1}} and {{math|1=''β'' = ''n'' &minus; ''k'' + 1}}, the beta distribution and the binomial distribution are related by{{clarification needed|date=March 2023|
reason=Is the left hand side referring to a probability density, and the right hand side to a probability mass function? Clearly a beta distributed random variable can not be a scalar multiple of a binomial random variable given that the former is continuous and the latter discrete. In any case, it would seem to be more correct to say that this relationship means that the PDF of one is related to the PMF of the other, rather than appearing to say that the _distributions_ (often interchangeable with their CDFs) are directly related to one another.
}} a factor of {{math|''n'' + 1}}:
: <math>\operatorname{Beta}(p;\alpha;\beta) = (n+1)B(k;n;p)</math>

[[Beta distribution]]s also provide a family of [[prior distribution|prior probability distribution]]s for binomial distributions in [[Bayesian inference]]:<ref name=MacKay>{{cite book| last=MacKay| first=David| title = Information Theory, Inference and Learning Algorithms|year=2003| publisher=Cambridge University Press; First Edition |isbn=978-0521642989}}</ref> 
: <math>P(p;\alpha,\beta) = \frac{p^{\alpha-1}(1-p)^{\beta-1}}{\operatorname{Beta}(\alpha,\beta)}.</math>
Given a uniform prior, the posterior distribution for the probability of success {{mvar|p}} given {{mvar|n}} independent events with {{mvar|k}} observed successes is a beta distribution.<ref>{{Cite web|url=https://www.statlect.com/probability-distributions/beta-distribution|title = Beta distribution}}</ref>