Date: Fri, 12 Aug 2022 05:10:50 +0000 (UTC) Message-ID: <1791061391.5710.1660281050737@localhost> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_5709_399800610.1660281050736" ------=_Part_5709_399800610.1660281050736 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Estimation of the probability p after having observed s successe= s in n trials

# Estimation of the probability p after having observed s successes i= n n trials

#### The idea

The results for the Binomial and Negative Binomial distribu= tions are both modeling randomness: that is to say th= at they are returning probability distributions of possible future outcomes= . At times, however, we are looking back at the results of a binomial proce= ss and wish to determine one of the parameters. For example, we may have ob= served n trials of which s were successes and from that i= nformation would like to estimate p. This binomial probability is = a fundamental property of the stochastic system and can never be observed, = but we can become progressively more certain about its true value by collec= ting data.

#### modeling the uncertainty about p

##### Bayesian statistics

If we have observed s su= ccesses in n random trials, a Bayesia= n analysis gives the conveniently simple result:

p=3DBeta(s+a, n-s+b, 1)<= /p>

where a Beta(a,b,1) prio= r is assumed.

The Beta(1,1,1) is a Uni= form(0,1) distribution - often considered appropriate as an uninformed prior, in which case we have:

p=3DBeta(s+1, n-s+1,1)

The Beta distribution can be used this way because it i= s a conjugate distribution to the binomial likeliho= od function.

##### Classical statistics

Three methods for the es= timation of p are discussed here.

##### Comparison between estimation methods<= /h5> A comparison of the clas= sical and Bayesian methods of estimating p is provided here.
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