Discrete random variable

Binomial distribution

A binomial distribution is the sum of independent Bernoulli trials with probability of success . #m/def/prob

The probability of a given value of is hence given by

and the Expectation and variance are given by

As , the shape of a binomial distribution approaches that of the continuous Normal distribution. See binomial coëfficient.

Properties

Let and let

  1. Expectation:
  2. Variance:
  3. Moment-generating function:
  4. Probability generating function:
Proof of 1–4

These follow from the analogous results for a Bernoulli trial by linearity, ^P3 and ^P1, since a binomial variable may be thought of as the sum of independent and identically distributed Bernoulli trials. ^P4 follows directly from the Binomial expansion.

Some further properties

  2. if is independent from

Relationship to other distributions

#state/tidy | #SemBr | #lang/en