# random variables

(1.7 hours to learn)

## Summary

Random variables are the central object of probability theory. As their name implies, can be thought of as variables whose values are randomly determined. Mathematically, they are represented as functions on a sample space.

## Context

This concept has the prerequisites:

- probability (Random variables are defined in terms of probability distributions.)

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

## -Free-

→ Khan Academy: Probability and Statistics

- Lecture "Random variables"
- Lecture "Discrete and continuous random variables"
- Lecture "Probability density functions"

→ Mathematical Monk: Probability Primer (2011)

Online videos on probability theory.

Other notes:

- This uses the measure theoretic notion of probability, but should still be accessible without that background. Refer to Lecture 1.S for unfamiliar terms.

## -Paid-

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

- Section 2.1, "Discrete random variables," up through 2.1.1, "Bernoulli random variables," pages 35-38
- Section 2.2, "Continuous random variables," not including subsections, pages 47-49

→ A First Course in Probability

An introductory probability textbook.

- Section 4.1, "Random variables," pages 132-138
- Section 4.2, "Discrete random variables," pages 138-140
- Section 5.1, "Introduction," pages 205-209

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

- Section 3.1, "Random variables and discrete distributions," pages 97-102
- Section 3.2, "Continuous distributions," pages 103-108

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

## -Free-

→ Sets, Counting, and Probability

Online lectures on basic probability theory.

Location:
Lecture "Random variables"

## -Paid-

→ An Introduction to Probability Theory and its Applications

A classic introductory probability textbook.

Location:
Section 9.1, "Random variables," pages 199-207

Additional dependencies:

- conditional probability

## See also

- Some important properties of random variables:
- expected value , the value it takes "on average"
- independence , i.e. one variable's distribution not depending on the other
- variance , how much the value tends to deviate from the mean
- entropy , a measure of the amount of uncertainty

- the binomial distribution , which counts the number of times an event occurs between identical and independent trials
- the multinomial distribution , where it takes values from a discrete set, each with some probability
- the Poisson distribution , used for modeling numbers of events that happen independently, such as earthquakes
- the exponential distribution
- the Gaussian distribution , a "bell curve" which appears extremely frequently in nature and is widely used in statistical modeling