Discrete random variable

Binomial distribution

A binomial distribution is the sum of independent Bernoulli trials with probability of success . 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 ^P2, ^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

• Relationship between binomial and hypergeometric distributions
• By the Central limits theorem, as .

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