18.600 | Fall 2019 | Undergraduate

# Probability and Random Variables

## Lecture Notes

SES # TOPICS
1 Permutations and combinations (PDF)
2 Multinomial coefficients and more counting (PDF)
3 Sample spaces and set theory (PDF)
4 Axioms of probability (PDF)
5 Probability and equal likelihood (PDF)
6 Conditional probabilities (PDF)
7 Bayes’ formula and independent events (PDF)
8 Discrete random variables (PDF)
9 Expectations of discrete random variables (PDF)
10 Variance (PDF)
11 Binomial random variables, repeated trials and the so-called modern portfolio theory (PDF)
12 Poisson random variables (PDF)
13 Poisson processes (PDF)
14 More discrete random variables (PDF)
15 Review for midterm exam 1 (PDF)
16 Midterm exam 1 [no lecture notes]
17 Continuous random variables (PDF)
18 Normal random variables (PDF)
19 Exponential random variables (PDF)
20 More continuous random variables (PDF)
21 Joint distribution functions (PDF)
22 Sums of independent random variables (PDF)
23 Expectation of sums (PDF)
24 Covariance and some conditional expectation exercises (PDF)
25 Conditional expectation (PDF)
26 Moment generating functions (PDF)
27 Weak law of large numbers (PDF)
28 Review for midterm exam 2 (PDF)
29 Midterm exam 2 [no lecture notes]
30 Central limit theorem (PDF)
31 Strong law of large numbers and Jensen’s inequality (PDF)
32 Markov chains (PDF)
33 Entropy (PDF)
34 Martingales and the optional stopping time theorem (PDF)
35 Martingales and risk neutral probability (PDF)
36 Risk neutral probability and Black-Scholes (PDF)
37 Review for final exam I (PDF)
38 Review for final exam II (PDF)
39 Review for final exam III (PDF)

Martingales, risk neutral probability, and Black-Scholes option pricing (PDF)—supplementary lecture notes for 34 to 36 which follow the outline of the lecture slides and cover martingales, risk neutral probability, and Black-Scholes option pricing (topics that do not appear in the textbook, but that are part of this course).

## Course Info

Fall 2019
##### Learning Resource Types
Problem Sets
Exams with Solutions
Lecture Notes
Online Textbook
Instructor Insights