UCLA MATH 170b | Practice Exams, Lecture Notes, Textbooks, Study Guides and Study Materials

problem_set_2

UCLA MATH 170b

Math 170B Section 1 Winter 2011 Problem Set # 2 Problem (1) Let X and Y denote two nonnegative real numbers not necessarily random variables. Show that 1 maxcfw_X, Y = lim (X N + Y N ) N . N Hint: Consider first the case X = Y and assume, without loss o
problem_set_3

UCLA MATH 170b

Math 170B Section 1 Winter 2011 Problem Set # 3 Be sure to put your section number on the cover of your problem set. Problem (1) Let X and Y denote the trinomial random variables fX,Y (n, m) = N pn q m (1 - (p + q)N -(n+m) . n, m On the basis of various p
problem_set_4

UCLA MATH 170b

Math 170B Section 1 Winter 2011 Problem Set # 4 Be sure to put your section number 0 P(|X| > a) in two ways: (i) Deduce it from Markov\'s inequality. (
problem_set_6

UCLA MATH 170b

Math 170B Section 1 Winter 2011 Problem Set # 6 Preliminary Posting Be sure to put your section number & TA name on the cover of your problem set. Problem (1) Recall the definition: A positive integer valued random variable T is a stopping time with respe
problem_set_5

UCLA MATH 170b

Math 170B Section 1 Winter 2011 Problem Set # 5 Be sure to put your section number & TA name on the cover of your problem set. Problem (1) Let X be a random variable with mean (with = 0) and variance 2 . In various applications, e.g., signal processing, o
math 170b sol

UCLA MATH 170b

Introduction to Probability 2nd Edition Problem Solutions (last updated: 11/25/11) c Dimitri P. Bertsekas and John N. Tsitsiklis Massachusetts Institute of Technology WWW site for book information and orders http:/www.athenasc.com Athena Scientific, Belmo
problem_set_2

UCLA MATH 170b

Math 170B Section 2 Winter 2013 Problem Set # 2 Problem (1) A random variable, X has a uniform distribution on [-1, 2]. Consider the function v(x) = -bx ax if x 0 if x 0. With a b > 0. Now let V be the random variable given by V = v(X). Find the pdf for V
problem_set_3

UCLA MATH 170b

Math 170B Section 2 Winter 2013 Problem Set # 3 Preliminary Posting Problem (1) Suppose X1 , X2 , . . . are independent Bernoulli random variables with parameter p, and let N = mincfw_i : Xi = 1 be the time of the first 1. Compute E(N ) and Var(N ) by con
problem_set_4

UCLA MATH 170b

Math 170B Section 2 Winter 2013 Problem Set # 4 Preliminary Posting Problem (1) Let Y be a binomial random variable with parameters N and p and X = x1 + + xN a sum of N independent Bernoulli random variables each of which have parameter p. Show that X and
homework1

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi c c Homework 1 From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end of the Chapter 4. And also the problems below: Problem 1. Give examples of (not independent) random variab
homework1-solutions

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi c c Homework 1 From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end of the Chapter 4. And also the problems below: Problem 1. Give examples of (not independent) random variab
homework2

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 2 c c From the textbook solve the problems 17, 18, 19, 22, 23 and 24 from the Chapter 4. Solve the problems 21, 22, 24, 30 from the Chapter 4 additional exercises at http:/www.athenasc.c
homework2-solutions

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 2 c c From the textbook solve the problems 17, 18, 19, 22, 23 and 24 from the Chapter 4. Solve the problems 21, 22, 24, 30 from the Chapter 4 additional exercises at http:/www.athenasc.c
homework3

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c 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
homework3-solutions

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c 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
homework4

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4 c c From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4. Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7 additional exe
homework4-solutions

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4 c c solutions From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4. Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7 addi
homework5

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 5 c c From the textbook solve the problems 4 and 5 from the Chapter 5. Solve the problems 2, 4, 6, 7, 9 and 10 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp
homework5-solutions

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 5 c c From the textbook solve the problems 4 and 5 from the Chapter 5. Solve the problems 2, 4, 6, 7, 9 and 10 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp
homework6

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 6 c c Solve the problems 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp.html And also the problems below: Problem 1. Denote points P0 ,
homework6-solutions

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 6 c c Solve the problems 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp.html And also the problems below: Problem 1. Denote points P0 ,
homework7

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 7 c c From the textbook solve the problems 8, 9, 10 and 11 from the Chapter 5. Solve the problems 11, 12, 13, 14 and 15 from the Chapter 7 additional exercises at http:/www.athenasc.com/
homework7-solutions

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 7 c c From the textbook solve the problems 8, 9, 10 and 11 from the Chapter 5. Solve the problems 11, 12, 13, 14 and 15 from the Chapter 7 additional exercises at http:/www.athenasc.com/
homework8

UCLA MATH 170b

Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 8 c c 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
midterm_1

UCLA MATH 170b

Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. If you have any questions raise your hand. The correct nal answer alone is not suc
midterm_1_practice_1

UCLA MATH 170b

Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) What is P()? (b) (2 points) If X and Y are independent random variables and va
midterm_1_practice_1-solutions

UCLA MATH 170b

Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) What is P()? Solution: P() = 0. (b) (2 points) If X and Y are independent rand
midterm_1_practice_2

UCLA MATH 170b

Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) If is a sample space, what is P(). (b) (2 points) If P(A) = 0.5, P(B ) = 0.4 a
midterm_1_practice_2-solutions

UCLA MATH 170b

Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) If is a sample space, what is P(). Solution: P() = 1 (b) (2 points) If P(A) =
midterm_1-solutions

UCLA MATH 170b

Midterm 1, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. If you have any questions raise your hand. The correct nal answer alone is not sucient for
midterm_2

UCLA MATH 170b

Midterm 2, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c 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 n
midterm_2_practice

UCLA MATH 170b

Midterm 2 practice, Math 170b, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) If X and Y are independent what is cov(X, Y )? (b) (2 points) Does there exist a rando
midterm_2_practice-solutions

UCLA MATH 170b

Midterm 2 practice, Math 170b, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) If X and Y are independent what is cov(X, Y )? Solution: 0. (b) (2 points) Does there
midterm_2-solutions

UCLA MATH 170b

Midterm 2, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c 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 n
midterm_3

UCLA MATH 170b

Midterm 3, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c 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 n
midterm_3_practice

UCLA MATH 170b

Midterm 3 practice, Math 170b, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) Random variable X satises P(X = 1) = 1/3 and P(X = 4) = 2/3. Find MX (s), the transfor
midterm_3-solutions

UCLA MATH 170b

Midterm 3, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c 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 n
solution_4

UCLA MATH 170b

Solution 4 Sec2.3 2.3.2(a) Let 1 3 1 0 - 3 C = 1 A= -1 2 - 1 , B = 4 1 2 , 1 -2 2 4 , and D = - 2 0 3 Compute A(2B+3C), (AB)D, and A(BD). Ans.: 3 12 5 3 6 2 0 - 6 3 2B+3C= 8 2 4 + - 3 - 6 0 = 5 - 4 4 1 3 5 3 6 20 - 9 18 = A(2B+3C)= 2 - 1 5 - 4 4 5 10 8
170Bp10s01a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 1 Mar 29, 2010 Homework assignments: Math 170B Probability, Sec. 1 01. Let X be a normal RV with parameters , 2 , and a, b R. (a) Show that Y = aX + b is also a normal RV (with which parameters?
170Bp10s02a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 2 Apr 02, 2010 Homework assignments: Math 170B Probability, Sec. 1 10. X, Y have the joint PDF f (x, y) = xe-x(y+1) 1cfw_x,y>0 . Calculate (a) E eXY (1+Y )2 (b) E(X) xy (c) E(XY ) 1 x -x = dxxe
170Bp10s03a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 3 Apr 09, 2010 Homework assignments: Math 170B Probability, Sec. 1 22. Suppose there are two methods (A, B) for measuring the distance from the earth to the moon. A scientist using method A gets
170Bp10s04a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 4 Apr 16, 2010 Homework assignments: Math 170B Probability, Sec. 1 36. Give a combinatorial proof (i.e a proof by counting something in two different ways) of the hypergeometric theorem. Answer:
170Bp10s05a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 5 Apr 23, 2010 Homework assignments: Math 170B Probability, Sec. 1 45. Choose a number n completely at random from cfw_1, 2, 3, then choose a number k completely at random from cfw_1, . . . , n.
170Bp10s06a

UCLA MATH 170b

spring 10, sheet 6 Apr 30, 2010 Mathematics Department, UCLA T. Richthammer Homework assignments: Math 170B Probability, Sec. 1 58. Conditional covariance formula. (a) Show that Cov (X, Y ) = E(Cov (X, Y |Z ) + Cov (E(X |Z ), E(Y |Z ). (b) Let X1 , X2 be
170Bp10s07a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 7 May 07, 2010 Homework assignments: Math 170B Probability, Sec. 1 71. In the lecture we considered Nx := mincfw_n : X1 + . . . + Xn x, where X1 , X2 , . . . are * independent RVs with uniform d
170Bp10s08a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 8 May 14, 2010 Homework assignments: Math 170B Probability, Sec. 1 86. Let X1 , X2 be independent normal RVs with parameters , 2 , and let X and V the corresponding sample mean and sample varia
170Bp10s09a

UCLA MATH 170b

spring 10, sheet 9 May 21, 2010 Mathematics Department, UCLA T. Richthammer Homework assignments: Math 170B Probability, Sec. 1 1 97. Let Xn be normal with parameters 0, n . Show that for n we have Xn 0 (a) in probability (b) in distribution (Hint: for (b
170Bp10s10

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, sheet 10 May 28, 2010 Homework assignments: Math 170B Probability, Sec. 1 107. We have shown that given Sn+1 = t, S1 , ., Sn are uniformly distributed on n . Show t without calculation that this impli
170Bp10sm1a

UCLA MATH 170b

Mathematics Department, UCLA T. Richthammer spring 10, midterm 1 Apr 19, 2010 First midterm: Math 170B Probability, Sec. 1 1. (9 pts) In the world chess championship 2010 the players A (Anand) and T (Topalov) will play 12 games. Every player gets one poin
170Bp10sm2a

UCLA MATH 170b

spring 10, midterm 2 May 17, 2010 Mathematics Department, UCLA T. Richthammer Second midterm: Math 170B Probability, Sec. 1 1. (7 pts) Let X1 , . . . , X6 be hypergeometric RVs with parameters n, N1 , . . . , N6 . (a) What is (by denition) the joint range
math170bhw1

UCLA MATH 170b

Mathematics 170B HW1 Due Thursday, January 6, 2011. Problems 1, 2, 3, 4 on page 246. (Note: On problem 4, the PDF of X is general not the uniform from Example 3.14 in Chapter 3.) The following problems are from my 170A nal exam last Fall. They are the one
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eval-170b.1.04s.pdf

UCLA MATH 170b

http://www.math.ucla.edu/~biskup/PDFs/evals/eval-170b.1.04s.pdf
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170B.HW1.pdf

UCLA MATH 170b

170B Homework 1: Selected solutions Ben Shargel April 12, 2007 Stirzaker Problem 3.20 Our problem is this: Given a group of 2n players entered into two tennis tournaments, what is the probability that there are no rematches in the opening round of th

http://www.math.ucla.edu/~shargel/170B.HW1.pdf
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170B.HW3.pdf

UCLA MATH 170b

170B Homework 3: Selected solution Ben Shargel April 24, 2007 3a. Let Yi be independent Bernoulli (1/2) variables and dene Xn as follows: 2n if Y1 = Y2 = = Yn = 1, X= 0 otherwise If X = 0 with probability 1, then Xn X a.s. Proof: Given > 0, n P

http://www.math.ucla.edu/~shargel/170B.HW3.pdf