Binomial distribution



Format: Binomial(n, p)

 

 

Uses

The Binomial distribution models the number of successes from n independent trials where there is the same probability p of success in each trial (see Binomial process). The binomial distribution has an enormous number of uses. Beyond simple modeling of binomial processes, many other stochastic processes can be usefully reduced to a binomial process to resolve problems. For example:

Binomial process:

Number of life insurance holders who will claim in a given period;

Number of loan holders who will default in a certain period;

Number of false starts of a car in n attempts;

Number of faulty items in n from a production line;

Number of n people randomly selected from a population who will have 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 antibodies;

Approximation to a hypergeometric distribution

Comments

The Binomial distribution makes the assumption that the probability p does not change the more trials are performed.

Example: the number of faulty computer chips in a 2000 volume batch where there is a 2% probability that any one chip is faulty = Binomial (2000, 2%).

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 special case of the Binomial with n = 1 i.e.: Bernoulli(p) = Binomial(1, p) that is used to model risk events.

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

Zero-modified versions

When modeling or analyzing counting data, it is often desirable to modify the probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring". We can make two types of modifications to our distribution for this:

  • Zero-inflated model - we increase the probability of zero events.

  • Zero-truncated model - we entirely remove the probability of zero events occurring.

ModelRisk functions added to Microsoft Excel for the Binomial distribution

VoseBinomial generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseBinomialObject constructs a distribution object for this distribution.

VoseBinomialProb returns the probability mass or cumulative distribution function for this distribution.

VoseBinomialProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseBinomialFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseBinomialFitObject constructs a distribution object of this distribution fitted to data.

VoseBinomialFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for the Zero-Inflated Binomial distribution

VoseZIBinomial generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZIBinomialObject constructs a distribution object for this distribution.

VoseZIBinomialProb returns the probability mass or cumulative distribution function for this distribution.

VoseZIBinomialProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZIBinomialFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZIBinomialFitObject constructs a distribution object of this distribution fitted to data.

VoseZIBinomialFitP returns the parameters of this distribution fitted to data.

 

ModelRisk functions added to Microsoft Excel for the Zero-Truncated Binomial distribution

VoseZTBinomial generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseZTBinomialObject constructs a distribution object for this distribution.

VoseZTBinomialProb returns the probability mass or cumulative distribution function for this distribution.

VoseZTBinomialProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZTBinomialFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZTBinomialFitObject constructs a distribution object of this distribution fitted to data.

VoseZTBinomialFitP returns the parameters of this distribution fitted to data.

 

Binomial distribution equations

Zero-Inflated Binomial distribution equations

Zero-Truncated Binomial distribution equations

 

 

 

 

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