# binomial distribution

(1.6 hours to learn)

## Summary

The binomial distribution describes the number of successes in a fixed number of independent trials, each with the same success probability. When the number of trials is large, the distribution is approximately bell-shaped.

## Context

This concept has the prerequisites:

- random variables
- independent events (The binomial distribution is motivated in terms of independent coin flips.)
- expectation and variance (The binomial distribution is parameterized in terms of its mean.)

## Goals

- Know the PMF of the binomial distribution

- Interpret the distribution in terms of i.i.d. Bernoulli random variables

- Describe the shape of the distribution

- Derive the expectation and variance of a binomial random variable

## Core resources (read/watch one of the following)

## -Free-

→ Khan Academy: Probability and Statistics

## -Paid-

→ A First Course in Probability

An introductory probability textbook.

Location:
Section 4.6, "The Bernoulli and binomial random variables," pages 150-160

→ An Introduction to Probability Theory and its Applications

A classic introductory probability textbook.

- Section 6.1, "Bernoulli trials," pages 135-136
- Section 6.2, "The binomial distribution," pages 136-139
- Section 6.3, "The central term and the tails," pages 139-141

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ Mathematical Monk: Probability Primer (2011)

Online videos on probability theory.

## -Paid-

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

Location:
Section 2.1.2, "The binomial distribution," pages 38-40

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 5.2, "The Bernoulli and binomial distributions," pages 247-250

Other notes:

- You can skip the MGF as far as this node is concerned.

## See also

- The multinomial distribution is the analog of the binomial distribution where each event can take more than two values.
- The probability parameter can be estimated from data using maximum likelihood .
- The Poisson distribution is the limiting case as the number of trials goes to infinty and the success probability goes to zero.
- When the number of trials is large, the binomial distribution is well approximated by the Gaussian distribution. This follows from the Central Limit Theorem .