Overview
Up until now, we have focussed on Point Estimates, i.e. using the data from a sample to give one predicted value for an unknown parameter. On this page, we study the first of two fundamental theorems necessary for the method of Interval Estimates which can add a measure of quality of our estimates: The Law of Large Numbers. At the same time, we also discuss a common misunderstanding of this theorem, the socalled Gambler’s Fallacy.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

Calculate the expected value and the variance of the sample mean from a simple random sample of size \(n\), when it is used to estimate the population mean (of any distribution!). (You’ll need to use the properties of expectation and variance of a sum of independent random variables.)

Know and understand the statement of the Weak Law of Large Numbers.

Understand the Gambler’s Fallacy, and the Inverse Gambler’s Fallacy.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Understand the proof of the Weak Law of Large Numbers, which is based on the result above and Chebyshev’s inequality.

Know the statement of the Strong Law of Large Numbers, a subtly different statement of the same principle which is sometimes easier to apply in practice.
To prepare for class

Watch the following video by Jeremy Jones which explains the intuitive idea behind the Law of Large Numbers and the Gambler’s Fallacy:

Watch the following video by MIT OpenCourseWare which explains & proves the Weak Law of Large Numbers:
After class

(Optional:) Watch the following video (by Harvard University) which goes through the detailed proofs of both the Weak and the Strong Law of Large Numbers, as well as the Central Limit Theorem (which we will discuss next):

(Optional:) Watch the following interesting video series by Kevin deLaplante about the Gambler’s Fallacy and its underlying concepts: