Date: Mon, 5 Jun 2023 23:31:48 +0000 (UTC) Message-ID: <653081634.604.1686007908491@localhost> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_603_1276392855.1686007908490" ------=_Part_603_1276392855.1686007908490 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Binomial

# Binomial

Binomial(p,n)

Binomial Equations

C= rystal Ball parameter restrictions

A Binomial(p,n) distribution returns discrete values between 0 and n. Ex= amples of the Binomial distribution are shown below: ### Uses

The Binomial distribution models the number of successes from n independ= ent trials where there is a probability p of success in each trial (as expl= ained in the section on the Binomial process).

The binomial distribution has an enormous number of uses. Beyond simple = binomial processes<= /a>, many other stochastic processes can be usefully reduced to a binomial = process to resolve problems. For example:

Binomial process:

Number of false starts of a car in n atte= mpts;

Number of faulty items in n from a produc= tion line;

Number of n randomly selected people with= some characteristic;

Reduced to binomial:

Number of machines that last longer than = T hours of operation without failure;

Blood samples that have zero, or >0 an= tibodies;

Approximation to a hypergeometric distribution

The following links lead to just some of t= he examples and models in ModelAssist that use the binomial distribution:

Conditional logic

Sampling from a liquid

Distribution fitting of threshold data

Bayesian prior

Test resu= lt

The Binomial distribution makes the assumption that the probability p do= es not change the more trials are performed. That would imply that my aim d= oesn't get better or worse. It wouldn't be a good estimator, for instance, = if the chance of success improved with the number of trials.

Another example: the number of faulty computer chips in a 2000 volume ba= tch where there is a 2% probability that any one chip is faulty =3D Binomia= l (2%,2000).

The Binomial distribution was first discussed by Bernoulli (1713). It is= related to the Beta and Negative Binomial distributions, all of which = have their basis in the Binomial process where the Binomial distribution is also derived. = The Bernoulli distribution is a speci= al case of the Binomial with n =3D 1 i.e.: Bernoulli(p) =3D Binomial(p,1) t= hat is used to model risk events.

The Binomial distribution has the property Binomial(p, n) + Binomial(p, = m) =3D Binomial(p,n+m) which makes sense if one thinks of n and m being two= sets of independent binomial trials, all with the same probability of succ= ess.

The Excel function BINOMDIST(s,n,p,0) returns the binom= ial probability mass function, and BINOMDIST(s,n,p,1) returns = the binomial cumulative distribution function.