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### The binomial distribution

The binomial distribution  is the discrete probability distribution applicable to the number of successes in a sequence of  independent yes/no experiments each of which has a success probability of . Each individual success/failure experiment is called a Bernoulli trial, so if  then the binomial distribution is a Bernoulli distribution.

It has the following characteristics:

 Distribution name Binomial distribution Common notation Parameters = number of (independent) trials, positive integer  = probability of success in each trial, Support Probability mass function Cumulative distribution function Mean Variance Skewness (Excess) kurtosis Characteristic function Other comments The Bernoulli distribution is  and corresponds to the likelihood of success of a single experiment.  Its probability mass function and cumulative distribution function are:   The Bernoulli distribution with , i.e. , has the minimum possible excess kurtosis, i.e. .   The mode of  is  if  is 0 or not an integer and is  if . If  then the distribution is bi-modal, with modes  and .

The binomial distribution is often used to model the number of successes in a sample size of  from a population size of . Since such samples are not independent, the resulting distribution is actually a hypergeometric distribution and not a binomial distribution. However if  then the binomial distribution becomes a good approximation to the relevant hypergeometric distribution and is thus often used.

In the above  is the binomial coefficient.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “binomial”. For details of other supported probability distributions see here.