Introduction to Probability
2nd Edition
Problem Solutions
(last updated: 9/18/15)
c
Dimitri P. Bertsekas and John N. Tsitsiklis
Massachusetts Institute of Technology
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Athena Scientific, Belmon
Lecture 1: Sets and Probabilistic Models
Hyang-Won Lee
Dept. of Internet & Multimedia Eng.
Konkuk University
Lecture 1
Probability and Statistics, Spring 2016
1 / 16
Probability
In non-mathematical sense, probability is interpreted in several
dierent ways
Probability and Statistics, S2016
Posted: March 23, 2016
Problem Set 2
Lecturer: Hyang-Won Lee
Due: March 30, 2015
1. Consider tossing a coin 100 times. Assume that each toss is independent of others. In each
toss, a head comes up with probability p.
(a)
Lecture 2: Conditional Probability
Hyang-Won Lee
Dept. of Internet & Multimedia Eng.
Konkuk University
Lecture 2
Probability and Statistics, Spring 2016
1/9
Conditional Probability
In many probabilistic experiments, knowledge of some event may
affect the
Lecture 4: Independence
Hyang-Won Lee
Dept. of Internet & Multimedia Eng.
Konkuk University
Lecture 4
Probability and Statistics, Spring 2016
1/1
Independence
Independent events
P(A|B) captures the partial information that event B provides about
event A
I
Lecture 6: Discrete Random Variables
Hyang-Won Lee
Dept. of Internet & Multimedia Eng.
Konkuk University
Lecture 6
Probability and Statistics, Spring 2016
1 / 11
Random Variables
Type of outcomes
Numerical values: die roll
Not numerical: coin toss (H/T)
I
Probability and Statistics, S2016
Posted: March 9, 2016
Problem Set 1
Lecturer: Hyang-Won Lee
Due: March 16, 2015
1. Use only the three probability axioms (and set algebra) to prove that for any event A,
P(Ac ) = 1 P(A).
2. You are going to meet your frie
Lecture 3: Total Probability Theorem and Bayes Rule
Hyang-Won Lee
Dept. of Internet & Multimedia Eng.
Konkuk University
Lecture 3
Probability and Statistics, Spring 2016
1/6
Total Probability Theorem
Total Probability Theorem
Let A1 , An be a partition of
Lecture 7: Mean, Variance and Joint PMFs
Hyang-Won Lee
Dept. of Internet & Multimedia Eng.
Konkuk University
Lecture 7
Probability and Statistics, Spring 2016
1 / 11
Expectation
It is sometimes desirable to summarize PMF in a single representative
number
Lecture 5: Counting
Hyang-Won Lee
Dept. of Internet & Multimedia Eng.
Konuk Univresity
Lecture 5
Prboaility and Statistisc , Spring 2016
1/8
Counting
Calculation of probabilities often involves counting outcomes
(a) All outcomes in are equally likely
P(A)