The binomial distribution
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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 bimodal, 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.
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