(1.6 hours to learn)
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.
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.)
- 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)
→ Khan Academy: Probability and Statistics
→ 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)
→ Mathematical Monk: Probability Primer (2011)
Online videos on probability theory.
→ 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
- You can skip the MGF as far as this node is concerned.
- 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 .
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