For an alternative approach, note that the event of interest occurs if and only if the time Y2 of the second arrival is less than or equal to 2. Hence, the desired probability is
2 2
P(Y2 2) =
0
fY2 (y ) dy =
(0.6)2 ye0.6y dy.
0
This integral can be evalu
Probability Theory, Math 170A - Homework 2
From the textbook solve the problems 14, 16, and 19 at the end of the
Chapter 1.
Solve the problems 15, 16, 18, 31, from the Chapter 1 additional exercises
at
http:/www.athenasc.com/prob-supp.html
And also the pr
Probability Theory, Math 170B, Spring 2013
Note: Although solutions exist on-line, you will be doing yourself a great
favor by resorting to them only after you have solved the problem yourself
(or at least tried very hard to).
From the textbook solve prob
Probability Theory, Math 170b - Homework 4
Problem 1. Show that for random variables X, Y and Z we have
E[E[E[X|Y ]|Z] = E[X].
Apply this formula to the following problem: Roll a far 6-sided die and observe the number Z
that came up. Then toss a fair coin
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
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Homework 3 solutions
From the textbook solve the problems 17, 18 and 19 at the end of the
Chapter 4.
From the books supplementary problems, solve problem 30 in Chapter 4
(see http:/www.athenasc
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
c
Homework 3
Due Friday, April 17th
From the textbook solve the problems 17, 18 and 19 at the end of the
Chapter 4.
From the books supplementary problems, solve problem 30 in Chapter 4
(see http:
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
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Homework 6
Due Friday, May 8th
From the books supplementary problems, solve problems 5, 6, 7, 9 and 19
in Chapter 7 (see http:/www.athenasc.com/prob-supp.html).
And also the problems below:
Pro
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
c
Homework 5 Due Friday, May 1st
From the books supplementary problems, solve problems 21, 22, 27 and 28
in Chapter 4, as well as 1, 3 and 4 in Chapter 7 (see http:/www.athenasc.com/probsupp.html
Probability Theory, Math 170b - Homework 5
From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4.
Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at
http:/www.athenasc.com/prob-supp.html
And also the pro
Probability Theory, Math 170A, Fall 2014 - Homework 5 solutions
From the textbook solve the problems 32, 39, 40 at the end of the Chapter 2.
Solution to Problem 32: Let Xi be the indicator of the event that the rst person in the
i-th couple is alive and Y
Probability Theory, Math 170A - Homework 4
From the textbook solve the problems 16, 22, 24 at the end of the Chapter 2.
Solve the problems 5 and 13 from the Chapter 2 additional exercises at
http:/www.athenasc.com/prob-supp.html
Problem 1. Recall Problem
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
c
Homework 6, solutions
Due Friday, May 8th
From the books supplementary problems, solve problems 5, 6, 7, 9 and 19 in Chapter 7 (see
http:/www.athenasc.com/prob-supp.html).
And also the problems
Probability Theory, Math 170b, Winter 2015 - Homework 7 solutions
From the textbook solve the problems 1, 2 and 3 from the Chapter 6.
Solve the problems 3, 4, 5, 6, 7, 8 and 9 from the Chapter 5 additional exercises at
http:/www.athenasc.com/prob-supp.htm
Probability Theory, Math 170a, Fall 2014- Homework 3 solution
From the textbook solve the problems 30, 33, 34, 35 and 36 at the end of
the Chapter 1.
Solution to Problem 30: In the rst case the hunter could choose the
correct path either if both dogs choo
Probability Theory, Math 170a, Fall 2014 - Homework 1 solutions
Problem 1. Show that for any sets A and B
P(A B) P(A) P(A B).
Solution: One way to solve it is to notice that A B A and A A B and use the properties
of the probability law (in particular the
Probability Theory, Math 170a, Fall 2014 - Homework 2 solutions
Problem 1. A person places randomly n letters in to n envelops . What is the probability that
exactly k letters reach their destination.
Solution See notes by Kupferman page 13.
Problem 2. Yo
Probability Theory, Math 170a, Winter 2015 - Homework 1
From the textbook solve the problems 2, 5-10 at the end of the Chapter 1.
And also the problems below:
Problem 1. Show that for any sets A and B
P(A B) P(A) P(A B).
Problem 2. We have a very weird di
Midterm 2, Math 170B - Practice 1
Printed name:
Signed name:
Student ID number:
Instructions:
Read problems very carefully. Please raise your hand if you have questions at any time.
The correct nal answer alone is not su cient for full credit - try to e
Midterm 2, Math 170b - Practice 2
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
Total:
40
Score
1. (a) (2 points) You roll a fair die N times where N is Poisson with parameter 1. What is the expected
value of the sum of the outcomes?
(b) (2 poi
Midterm 2 practice, Math 170b, Spring 2015
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
Total:
40
Score
1. (a) (2 points) Can a random variable have the same PDF and CDF? Justify your reasoning.
(b) (2 points) Does there exist a random variabl
Midterm 2, Math 170B - Practice 1 solutions
Printed name:
Signed name:
Student ID number:
Instructions:
Read problems very carefully. Please raise your hand if you have questions at any time.
The correct nal answer alone is not sucient for full credit -
Probability Theory, Math 170b, Spring 2015, Homework 7
due Friday, May 15th
Solve the problems 10, 11, 13, 15, 17 and 18 from the Chapter 7 additional exercises at
http:/www.athenasc.com/prob-supp.html
And also the problems below:
Problem 1. Let X1 , X2 ,
Probability Theory, Math 170A - Homework 5
From the textbook solve the problems 1 and 2 at the end of the Chapter 3.
Solve the problems 1 and 2 from the Chapter 3 additional exercises at
http:/www.athenasc.com/prob-supp.html
And also the problems below
Pr
EXERCISES
1. Verify that the derivatives of sinh z and cosh z are as stated in equations (2), Sec. 34.
2. Prove that sinh 22 = 2 sinh z cosh z by starting with
( a ) definitions (I), Sec. 34, of sinh z and cosh z ;
(b) the identity sin 22 = 2 sin z cos z
Probability Theory, Math 170A - Homework 7
Problem 1. Is it always the case that lim supn An is not the empty set?
Problem 2. Find a sequence of events which does not have a limit.
Problem 3. Prove that if (An ) is a decreasing sequence of events, then it
Probability Theory, Math 170b, Spring 2015, Homework 7
due Friday, May 15th
Solve the problems 10, 11, 13, 15, 17 and 18 from the Chapter 7 additional exercises at
http:/www.athenasc.com/prob-supp.html
And also the problems below:
Problem 1. Let X1 , X2 ,
Midterm 2, Math 170b - Practice 2 solutions
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
Total:
40
Score
1. (a) (2 points) You roll a fair die N times where N is Poisson with parameter 1. What is the expected
value of the sum of the outcomes?
Introduction to Probability
2nd Edition
Problem Solutions
(last updated: 10/22/13)
c
Dimitri P. Bertsekas and John N. Tsitsiklis
Massachusetts Institute of Technology
WWW site for book information and orders
http:/www.athenasc.com
Athena Scientic, Belmont
CHAP. 4
Thus, when z lies on C,
Writing M = 6/7 and observing that L = K is the length of C , we may now use
inequality (1) to obtain inequality (2).
EXAMPLE 2. Here C R is the semicircular path
z = Re i 6
and z
(0 59 5 ~ ) ,
denotes the branch
of the squ
ANTIDERIVATIVES 137
SEC.42
statement (iii), equation (3) holds (see Fig. 48). Thus equation (2) holds. Integration
is, therefore, independent of path in D ; and we can define the function
on D. The proof of the theorem is complete once we show that F'(z)