# moment generating functions

(1.3 hours to learn)

## Summary

The moment generating function (MGF) is a function which characterizes the distribution of a random variable. MGFs are useful for analyzing sums of independent random variables. In particular, they are used in the proof of the Central Limit Theorem and in deriving the Chernoff bounds, which bound the probability that a sum of independent random variables takes on extreme values.

## Context

This concept has the prerequisites:

- expectation and variance (The MGF is closely related to moments of a distribution.)
- independent random variables (MGFs are a powerful tool for analyzing sums of independent random variables.)

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

## -Paid-

→ A First Course in Probability

An introductory probability textbook.

Location:
Section 7.7. "Moment generating functions," not counting 7.7.1, pages 387-397

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 4.4, "Moments," pages 203-208

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

Location:
Section 4.5, "The moment-generating function," pages 155-161

## See also

- MGFs are an important tool for proving several important results:
- central limit theorem , which characterizes the limiting distribution of the mean of i.i.d. random variables
- Chernoff bounds , which bound the probability of sums of random variables taking extreme values