Binomial Distribution
The binomial distribution is a discrete probability density function.
Description
The distribution describes the sum of repeated and independent Bernoulli-distributed events.
The probability of k successes can be thought of as the number of ways k can be selected from n Bernoulli trials. Naturally the density function is parameterized in terms of n, k, and p.
for integer values of k ∈ [0,n].
It follows that the cumulative density function is:
Moments
The expected value is given as E[X] = np.
Variance is given as Var[X] = np(1 - p) = npq.
Usage
Sum of Repeated Bernoulli Trials
The sum of repeated (independent) Bernoulli trials is known to follow a binomial distribution. In other words, if X ~ Bernoulli(p) then nX̅ ~ Binomial(n,p).