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 (
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
a
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
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
c
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
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 supplemen
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
c
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.athenas
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
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 o
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 even
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
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 addit
Probability Theory, Math 170a, Fall 2014 - Homework 7 solutions
From the textbook solve the problems 6, 7, 11 and 15 at the end of the Chapter 3.
Solution to Problem 6: Let X be her waiting time then
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:/
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
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 th
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
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
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.
Probability Theory, Math 170a, Fall 2014 - Homework 6 Solutions
From the textbook solve the problems 1 and 2 at the end of the Chapter 3.
Solution to Problem 1: The PMF of Y is
P(Y = 1) = P(X 1/3) = 1
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
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 reas
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 th
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 co
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.htm
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 th
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
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.htm
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.
CHAP.
4
But
(xOu' - y0v')
+ i(yOu'+ xOv')= (xo + iy0)(uf+ iv') = z0wr(t),
and so
d
dt
-[zow( t ) ]= zow'(t).
(3)
Another expected rule that we shall often use is
where zo = xo
+ iyo. To verify this, w
SEC.
COMPLEX
EXPONENTS
32
97
(6) showing that log(l/z) = - log z (z # 0), in the sense that log(l/z) and - log z have
the same set of values, and then referring to expression (I), Sec. 31, for log(zlz