Probability Theory – Week 4 Summary (Blitzstein & Hwang)
This document provides a broad summary of Week 4 of Probability Theory, aligned with the
corresponding chapters in Blitzstein & Hwang, Introduction to Probability.
1. Expectation (Chapter 4.1–4.2)
Expectation represents the long-run average value of a random variable. For discrete
random variables, it is defined as the weighted average of possible values. Linearity of
expectation allows expectations of sums and scaled variables to be computed easily,
without requiring independence. Expectation may fail to exist if infinite sums diverge.
2. Law of the Unconscious Statistician (LOTUS) (Chapter 4.3)
LOTUS provides a method to compute the expectation of a function of a random variable
directly from the distribution of the original variable. This avoids the need to determine the
distribution of the transformed variable and is widely used in moment calculations.
3. Indicator Random Variables (Chapter 4.4)
Indicator random variables take the value 1 if an event occurs and 0 otherwise. Their
expectation equals the probability of the event. This creates a fundamental bridge between
probability and expectation and enables powerful counting arguments using linearity of
expectation.
4. Variance and Standard Deviation (Chapter 4.5)
Variance measures the spread of a random variable around its mean. It can be computed
using Var(X) = E[X²] − (E[X])². Variance scales quadratically with constants and is additive
for independent random variables. Standard deviation provides a measure of spread in the
original units.
5. Poisson Distribution (Chapter 7.1–7.2)
The Poisson distribution models the number of rare events occurring over a fixed interval
with a known average rate. It arises as a limit of the binomial distribution when the number
of trials is large and the success probability is small. The mean and variance of a Poisson
random variable are both equal to its rate parameter λ. Independent Poisson variables add
to form another Poisson variable.
6. Negative Binomial Distribution (Chapter 7.4)
The Negative Binomial distribution models the number of failures before a fixed number of
successes occurs. It generalizes the geometric distribution and can be represented as a sum
of independent geometric random variables. Its expectation and variance follow naturally
from this representation.
Overall Takeaway
Week 4 emphasizes expectation and variance as central tools in probability theory. These
concepts simplify analysis, allow elegant solutions to complex problems, and underpin key
discrete distributions such as the Poisson and Negative Binomial.