MA/STAT 519: Introduction to Probability Theory
Spring 2018, Purdue University
http://www.math.purdue.edu/~yip/519
Course Description:

Algebra of sets, sample spaces, combinatorial problems, conditional
probability and independence,
discrete and continuous random variables, distribution and joint distribution functions, expectation of random variables, moment generating
functions, limit theorems and applications.
Instructor:
 Aaron Nung Kwan
Yip
 Department of
Mathematics
 Purdue University
Contact Information:
 Office: MATH 432
 Email:
click here
Lecture Time and Place:
 T, Th 10:30  11:45, UNIV 117
Office Hours:
 Tue, 4:305:30pm, Wed, 3:004:00pm, or by appointment.
Textbook:

(All of the following are either available online or on reserve in
the math library.)
Main Text:
[CD] Probability with Applications in Engineering, Science, and
Technology,
by Matthew A. Carlton, Jay L. Devore, 2nd edition
(available online from Purdue library page)
Reference:
[R] A First Course in Probability, by Sheldon Ross
(I am using the 9th edition,
but older editions are OK for reading and reference)
[HPS] Introduction to Probability Theory, by Hoel, Port and Stone
[D1] Fundamentals of Probability: a First Course,
by Anirban DasGupta (available online from Purdue library page)
[D2] Probability for Statistics and Machine Learning:
Fundamentals and Advanced Topics,
by Anirban DasGupta (available online from Purdue library page)
[F1, F2] An Introduction to Probability Theory and Its Applications,
Volume 1 and 2,
by William Feller
Course Outline:
 The course will cover most of the sections of
[CD] Chapters 1 to 4, and part of Chapter 5.
Prerequisites:

Good "working" knowledge of:
linear algebra (e.g. MA 265, 351, 511),
vector calculus (e.g. MA 261, 362, 510),
and mathematical analysis (e.g. MA 341, 440, 504).
Homework:

Homeworks will be assigned roughly biweekly.
They will be gradually assigned as the course progresses.
Please refer to the course announcement below.
 Steps must be shown to explain your answers.
No credit will be given for just writing down the answers, even
if it is correct.
 Please staple all loose sheets of your homework to prevent
5% penalty.
 Please resolve any error in the grading (hws and tests)
WINTHIN ONE WEEK after the return of each homework and exam.
 No late homeworks are accepted (in principle).
 You are encouraged to discuss the homework problems with
your classmates but all your handedin homeworks must be your
own work.
Examinations:
 Midterm Test: Date to be determined
 Final Exam: During Final Exam Week
Grading Policy:
 Homeworks (50%)
 Midterm Test (20%)
 Final Exam (30%)
You are expected to observe academic honesty
to the highest standard. Any form of cheating will automatically
lead to an F grade, plus any other disciplinary action,
deemed appropriate.
Accommodations for Students with Disabilities and
Academic Adjustment:
 University policy will be followed.
Course Progress and Announcement:
 (You should consult this section regularly,
for homework assignments, additional materials and announcements.)
WEEK 1
Jan 9 (Tue):
[CD 1.11.2][R 2.12.4][D1, Chapter 1]
Concept of statistical regularity,
Law of Large Numbers (LLN),
Central Limit Theorem (CLT),
Probability space, outcome, events, logical operations between events,
complement, unions, intersections, de Morgan's Law.
Jan 11 (Thur):
[CD 1.11.2][R 2.12.4][D1, Chapter 1]
Probability measure, countablyadditivity,
inclusionexclusion principle [R p. 30, Proposition 4.4][D1, 1.5],
objective vs subjective approaches to probability measures.
WEEK 2
Jan 16 (Tue):
Some theoretical remarks about probability measures:
countablyaddivitiy, (non)measureable sets, typical outcomes,
equallylikelyoutcome probability space;
[CD 1.3][R 1.11.4][D1 1.4]
counting techniques: permutations and combinations.
Jan 18 (Thur) [R 1.11.6]:
binomial and multinomial coefficients and expansions,
number of integer solutions.
Homework 1, due Thursday, Jan 25,
in class
WEEK 3
Jan 23 (Tue): [D1 6.5, Chapter 2][R 2.5][CD 1.3.3, 2.6.1]
Sampling (partition, hypergeometric distribition),
birthday and matching problems,
asymptotics of formula when some number
(e.g. population size) is large.
Jan 25 (Thur):
[D1 Chapter 15] Urn and occupancy problems,
MaxwellBoltzmann, BoseEinstein and FermiDirac Distributions;
[R 3.13.3][D1 Chapter 3][CD, 1.4]
Conditional probability,
Bayes' forward (formula of total probability) and
backward (updating prior) formula,
prior and posterior probabilities.
WEEK 4
Jan 30 (Tue):
(cont.) Bayes' backward and forward formula, prior and posterior
probabilities, sensitivity and specificity of medical
tests.
[R, 3.13.4][D1 Chapter 3][CD 1.5]
independence between two, three, and multiple events.
Feb 1 (Thur):
[R, 3.13.4][D1 Chapter 3][CD 1.5]
(cont.) (jointly) indepedence between multiple events.
"Probability paradoxes": Monty Hall
Homework 2, due Thursday, Feb 8, in class
WEEK 5
Feb 6 (Tue):
"Probability paradoxes": (Mr. Smith's) Boy vs Girl.
computation of probability by iteration/recursion/induction,
an example [R, e.g. 5d, p. 93]: matching problem.
Feb 8 (Thur):
iterations/recursion/induction, Gambler's Ruin Problem [R, p. 84],
trials with random (unknown) parameters,
conditional independence.
WEEK 6
Feb 13 (Tue):
[R 4.14.6][D1 Chapter 4][DC Chapter 2.12.4]
(Discrete) Random variables (numerical observations of experimental
outcomes),
probability mass (density) function (pmf, pdf) and
cumulative density function (cdf),
expectation, variance, CauchySchwarz Inequality,
Bernoulli, Binomial, and Poisson random variables.
Note on random variables
and expectation.
Feb 15 (Thur):
Analytical computation of expectations and
variance of Binomial and Poisson random variables
[R Chapter 4/6.26.3/7.27.3]
Joint (bivariate and multivariate) random variables,
joint pdf, marginal pdfs, independent random variables,
computations of Ef(X,Y), E(X+Y), E(XY)
Homework 3, due Thursday, Feb 22, in class
WEEK 10
Midterm, Mar 22, in class (review on Mar 20)