18.600 | Fall 2019 | Undergraduate

Probability and Random Variables


Readings are assigned from the required textbook.

Ross, Sheldon. A First Course in Probability. 8th ed. Pearson Prentice Hall, 2009. ISBN: 9780136033134.

Additional readings are available in the links listed below.

1 Permutations and combinations Sections 1.1–1.3 (also Pascal’s triangle, see also correspondence with Fermat: Fermat and Pascal on Probability (PDF))
2 Multinomial coefficients and more counting Sections 1.4–1.5 (see Pascal’s pyramid)
3 Sample spaces and set theory Sections 2.1–2.2
4 Axioms of probability Sections 2.3–2.4 (see Paulos’ NYT article and conjunction fallacy blog and famous hat problem)
5 Probability and equal likelihood Sections 2.5–2.7 (and a bit more history)
6 Conditional probabilities Sections 3.1–3.2 (and Conditional risk)
7 Bayes’ formula and independent events Sections 3.3–3.5
8 Discrete random variables Sections 4.1–4.2
9 Expectations of discrete random variables Sections 4.3–4.4 (and, for non-discrete setting, examples of non-measurable sets, as in the Vitali construction)
10 Variance Section 4.5
11 Binomial random variables, repeated trials and the so-called modern portfolio theory Section 4.6 (and the so-called modern portfolio theory)
12 Poisson random variables Section 4.7 (and Soccer goal probabilities: Poisson vs actual distribution)
13 Poisson processes Section 9.1
14 More discrete random variables Sections 4.8–4.9
15 Review for midterm exam 1 No Readings
16 Midterm exam 1 No Readings
17 Continuous random variables Sections 5.1–5.3
18 Normal random variables Section 5.4
19 Exponential random variables Section 5.5
20 More continuous random variables Sections 5.6–5.7
21 Joint distribution functions Sections 6.1–6.2
22 Sums of independent random variables Sections 6.3–6.5
23 Expectation of sums Sections 7.1-7.2
24 Covariance and some conditional expectation exercises Sections 7.3-7.4
25 Conditional expectation Sections 7.5–7.6
26 Moment generating functions Sections 7.7–7.8
27 Weak law of large numbers Sections 8.1–8.2
28 Review for midterm exam 2 No Readings
29 Midterm exam 2 No Readings
30 Central limit theorem Section 8.3
31 Strong law of large numbers and Jensen’s inequality Sections 8.4–8.5 (see also the truncation-based proof on Terry Tao’s blog and the characteristic function proof of the weak law)
32 Markov chains Section 9.2
33 Entropy Sections 9.3–9.4
34 Martingales and the optional stopping time theorem MartingalesOptional stopping time theorem, and Martingales, risk neutral probability, and Black-Scholes option pricing (PDF) (see also prediction market plots)
35 Martingales and risk neutral probability Martingales, risk neutral probability, and Black-Scholes option pricing (PDF)
36 Risk neutral probability and Black-Scholes Black-Scholes and Martingales, risk neutral probability, and Black-Scholes option pricing (PDF) (look up options quotes at the Chicago Board Options Exchange)
37 Review for final exam No Readings
38 Review for final exam (cont.) No Readings
39 Review for final exam (cont.) No Readings
40 Final exam No Readings

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As Taught In
Fall 2019
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