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MATH 170b - Probability Theory - UCLA Study Resources
  • 28 Pages homework3
    Homework3

    School: UCLA

    Course: Probability Theory

    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

  • 32 Pages homework1-solutions
    Homework1-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 38 Pages homework4-solutions
    Homework4-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 38 Pages midterm_1_practice_2-solutions
    Midterm_1_practice_2-solutions

    School: UCLA

    Course: Probability Theory

    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) =

  • 41 Pages midterm_3-solutions
    Midterm_3-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 41 Pages midterm_2-solutions
    Midterm_2-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 133 Pages math 170b sol
    Math 170b Sol

    School: UCLA

    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

  • 39 Pages homework3-solutions
    Homework3-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 2 Pages homework6-solutions winter
    Homework6-solutions Winter

    School: UCLA

    Course: Probability Theory

    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 ,

  • 33 Pages homework5
    Homework5

    School: UCLA

    Course: Probability Theory

    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

  • 36 Pages homework2-solutions
    Homework2-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 31 Pages homework4
    Homework4

    School: UCLA

    Course: Probability Theory

    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

  • 8 Pages PracticeFinal_SolsSec1
    PracticeFinal_SolsSec1

    School: UCLA

  • 2 Pages homework1
    Homework1

    School: UCLA

    Course: Probability Theory

    Math 170B, Spring 2015 - Homework 1 Due Friday April 3rd Problem 1. Let X be a Uniform random variable on [a, b], where 0 < a < b < . Given X = x, let Y be an exponential random variable with parameter x. Let t 0. P(Y > t|X = x). P(Y > t). Problem 2. Le

  • 2 Pages Midterm #2 Solutions
    Midterm #2 Solutions

    School: UCLA

    Course: Probability Theory

    T. Liggett Mathematics 170B Midterm 2 Solutions May 23, 2012 (20) 1. (a) State Markovs inequality. Solution: If X 0, then P (X a) EX/a for a > 0. (b) Prove Markovs inequality. Solution: a1cfw_X a X . Taking expected values gives aP (X a) EX . (c) Suppose

  • 4 Pages PracticeFinal
    PracticeFinal

    School: UCLA

    UCLA id: Name: Math 170b Winter 2012 Final Exam Section Points Grade I 35 II 32 III 33 Total 100 This is a practice nal. For the sake of time, I did not add more problems than can be solved. Expect the real nal to require solving the same amount of probl

  • 2 Pages Midterm #1 Solutions
    Midterm #1 Solutions

    School: UCLA

    Course: Probability Theory

    T. Liggett Mathematics 170B Midterm 1 Solutions April 25, 2012 (10) 1. Suppose X has the Cauchy density 1 , < x < . f (x) = (1 + x2 ) Find the density of Y = 2X + 1. Solution: y 1 2 y1 1 = dx, P (Y y ) = P X 2 2 (1 + x ) so Y has density 2 1 . 4 + (y

  • 9 Pages Homework Solutions
    Homework Solutions

    School: UCLA

    Course: Probability Theory

    Mathematics 170B Selected HW Solutions. F4 . Suppose Xn is B (n, p). (a) Find the moment generating function Mn (s) of (Xn np)/ np(1 p). Write q = 1 p. The MGF of Xn is (pes + q )n , since Xn can be written as the sum of n independent Bernoullis with para

  • 1 Page hwk1
    Hwk1

    School: UCLA

    Math 170B Winter 2014 Homework 1 Suggested reading: Sections 4.1, 4.2, and 4.3. Read the solutions to Problem 15, Problem 20 from the end of Chapter 4. Problems: (1) Solve problems 5, 18 and 19 from Chapter 3. (2) Solve problems 5, 7 and 14 from Chapter

  • 1 Page hwk3
    Hwk3

    School: UCLA

    Math 170B Winter 2014 Homework 3 Suggested reading: Section 5.1. Read the solutions to Problems 39, 45 from the end of Chapter 4. Problems: (1) Solve Problems 34, 35, 42 and 43 from Chapter 4. (2) Compute (a) X is (b) X is (c) X is (d) X is (e) X is a f

  • 1 Page hwk2
    Hwk2

    School: UCLA

    Math 170B Winter 2014 Homework 2 Suggested reading: Section 4.4. Read the solutions to Problems 21, 25, 26 and 27 from the end of Chapter 4. Problems: (1) Solve Problems 10, 11, 12, 29, 32 and 33 from Chapter 4. (2) A fair die is tossed n times, let X d

  • 1 Page hwk4
    Hwk4

    School: UCLA

    Math 170B Winter 2014 Homework 4 Suggested reading: Sections 5.2, 5.3 and 5.4. Read the solutions to Problem 2 (Cherno bound) from the end of Chapter 4. Problems: (1) Solve Problems 1, 4 and 5 from Chapter 5. (2) Let X1 , X2 , X3 , . . . be a sequence o

  • 1 Page hwk5
    Hwk5

    School: UCLA

    Math 170B Winter 2014 Homework 5 Suggested reading: Section 5.5. Solutions to problems 12, 18 at the end of chapter 5. Notes (posted on my homepage/CCLE) about the Berry-Essen Theorem. Problems: (1) Let X1 , X2 , . . . be i.i.d and uniformly distribute

  • 2 Pages hwk6
    Hwk6

    School: UCLA

    Math 170B Winter 2014 Homework 6 Suggested reading: Section 6.1 (note: it is a longer section than usual). Keep in mind that I will be departing at times from the book, adding some extra material during lectures. Notably, the Gamblers ruin example from

  • 2 Pages hwk7
    Hwk7

    School: UCLA

    Math 170B Winter 2014 Homework 7 Suggested reading: Section 6.2. Solutions to Problems 4, 19, 22. Problems: (1) Solve problems 12, 13 and 16 at the end of chapter 6. (2) We are given a biased coin for which the probability of heads is p (0 < p < 1). (a)

  • 1 Page hwk3
    Hwk3

    School: UCLA

    Math 170B Winter 2014 Homework 3 Suggested reading: Section 5.1. Read the solutions to Problems 39, 45 from the end of Chapter 4. Problems: (1) Solve Problems 34, 35, 42 and 43 from Chapter 4. (2) Compute (a) X is (b) X is (c) X is (d) X is (e) X is a f

  • 4 Pages BerryEsseen
    BerryEsseen

    School: UCLA

    The Berry-Esseen inequality Let X1 , X2 , . . . be an innite i.i.d. sequence of random variables. If and common mean and variance, dene Sn by p n X1 + . . . + Xn Sn = . n 2 denote their Then, the Berry-Esseen inequality is (x)| p |FSn (x) n 3 , (BE) whe

  • 2 Pages SolutionsHwk1
    SolutionsHwk1

    School: UCLA

  • 6 Pages SolutionsHwk2
    SolutionsHwk2

    School: UCLA

  • 2 Pages Homework 5
    Homework 5

    School: UCLA

    Course: Probability

    Probability Theory, Math 170B - Homework 5 From the textbook solve problems 29 , 30 , 32 , 33 , 36 , 38, 41 , 42, 43 in Chapter 4. Problems marked with a are not mandatory but highly recommended. Solve problems 1,4,6, 27,28 from Chapter 4 additional exerc

  • 5 Pages Homework 3 Solutions
    Homework 3 Solutions

    School: UCLA

    Course: Probability

    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

  • 2 Pages Homework 7
    Homework 7

    School: UCLA

    Course: Probability

    Probability Theory, Math 170B - Homework 7 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.html And also the problems

  • 2 Pages Homework 3
    Homework 3

    School: UCLA

    Course: Probability

    Probability Theory, Math 170B Exercise 3 From the textbook solve problems 8 and 14 in Section 4.1 and 17, 18, 19 in Section 4.2 (appear at the end of the chapter). From the books supplementary problems, solve problems 30, 31, 33 and 34 in Chapter 4 (see h

  • 1 Page Hoemwork 9 with Solutions
    Hoemwork 9 With Solutions

    School: UCLA

    Course: Probability

    Probability Theory, Math 170b, Winter 2014 - Homework 9 Do not hand in this homework From the textbook solve the problems 8, 12, 16 from the Chapter 6. Solve the problems 10, 16, 26, and 35 from the Chapter 5 additional exercises at http:/www.athenasc.com

  • 4 Pages Homework 7 Solutions
    Homework 7 Solutions

    School: UCLA

    Course: Probability

    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

  • 1 Page Homework 1
    Homework 1

    School: UCLA

    Course: Probability

    Probability Theory, Math 170B, Winter 2015 - Homework 1 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 ev

  • 1 Page Homework 8
    Homework 8

    School: UCLA

    Course: Probability

    Probability Theory, Math 170b, Spring 2014, - Homework 8 Problem 1. Let a and b be distinct fixed real numbers, such that a + b > 0 and a < b. Consider a random walk Xn starting from X0 = 0 and such that either Xn+1 Xn = a or Xn+1 Xn = b with equal probab

  • 1 Page Homework 2
    Homework 2

    School: UCLA

    Course: Probability

    Probability Theory, Math 170B, Homework 2 We will prove the convolution formula for the sum of independent random variables Z = X + Y : fZ (z) = fX (x)fY (z x)dx. From the textbook solve the problems 1, 2, 3, 5, 6, 8 and 11 at the end of the Chapter 4. Fr

  • 28 Pages Entire Class Notes Part 2
    Entire Class Notes Part 2

    School: UCLA

    Course: Probability

    Scanned by Ca

  • 4 Pages Midterm 1
    Midterm 1

    School: UCLA

    Course: Probability

  • 2 Pages Homework 4 Solutions
    Homework 4 Solutions

    School: UCLA

    Course: Probability

    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

  • 2 Pages Homework 2 Solutions
    Homework 2 Solutions

    School: UCLA

    Course: Probability

    Probability Theory, Math 170b - Homework 2 solutions Problem 1. Let X be exponentially distributed with parameter . Find the PMF of Y = X , where x for a real number x is the rounding of x to the nearest integer whose value is greater or equal to x. Ident

  • 2 Pages SolutionsHwk3
    SolutionsHwk3

    School: UCLA

  • 7 Pages SolutionsHwk6
    SolutionsHwk6

    School: UCLA

  • 1 Page homework2
    Homework2

    School: UCLA

    Course: Probability Theory

    Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 2 Due Friday, April 10th From the textbook solve the problems 1, 2, 3, 5, 6, 7 and 11 at the end of the Chapter 4. From the books supplementary problems, solve problem 19 in Chapter 3

  • 3 Pages homework1-solutions
    Homework1-solutions

    School: UCLA

    Course: Probability Theory

    Math 170B, Spring 2015 - Homework 1 Due Friday April 3rd Problem 1. Let X be a Uniform random variable on [a, b], where 0 < a < b < . Given X = x, let Y be an exponential random variable with parameter x. Let t 0. P(Y > t|X = x). P(Y > t). Solution: 1.

  • 1 Page homework3
    Homework3

    School: UCLA

    Course: Probability Theory

    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:

  • 2 Pages homework2-solutions
    Homework2-solutions

    School: UCLA

    Course: Probability Theory

    Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 2 Due Friday, April 10th From the textbook solve the problems 1, 2, 3, 5, 6, 7 and 11 at the end of the Chapter 4. From the books supplementary problems, solve problem 19 in Chapter 3

  • 6 Pages midterm_2_practice_1-solutions
    Midterm_2_practice_1-solutions

    School: UCLA

    Course: Probability Theory

    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. Solution: No. The CDF FX is always non-negative

  • 5 Pages midterm_2_practice-2
    Midterm_2_practice-2

    School: UCLA

    Course: Probability Theory

    Practice Midterm 2 version 2, Math 170b, Spring 2015 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

  • 5 Pages midterm_2_practice-2-solutions
    Midterm_2_practice-2-solutions

    School: UCLA

    Course: Probability Theory

    Practice Midterm 2 version 2, Math 170b, Spring 2015 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

  • 2 Pages Homework 5 Solutions
    Homework 5 Solutions

    School: UCLA

    Course: Probability

    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

  • 3 Pages Homework 6 Solutions
    Homework 6 Solutions

    School: UCLA

    Course: Probability

    Probability Theory, Math 170B - Homework 6 From the textbook solve the problems 4 and 5 from the Chapter 5. Solve the problems 6, 7, 9, 10 and 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp.html And also

  • 27 Pages Entire Class Notes Part 1
    Entire Class Notes Part 1

    School: UCLA

    Course: Probability

    Scanned by Ca

  • 6 Pages Final Exam Solutions
    Final Exam Solutions

    School: UCLA

    Course: Probability Theory

    T. Liggett Mathematics 170B Final Exam Solutions June 13, 2012 (16) 1. Let N (t) be a Poisson process with rate = 2, and for 0 a < b, let N (a, b) = N (b) N (a) be the number of Poisson points in the interval (a, b). (a) Find P (N (2, 3) = 6 | N (0, 5) =

  • 1 Page math170bhw1
    Math170bhw1

    School: UCLA

    Course: Probability

    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

  • 44 Pages midterm_1_practice_1-solutions
    Midterm_1_practice_1-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 29 Pages midterm_1_practice_2
    Midterm_1_practice_2

    School: UCLA

    Course: Probability Theory

    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

  • 41 Pages midterm_1-solutions
    Midterm_1-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 35 Pages midterm_2
    Midterm_2

    School: UCLA

    Course: Probability Theory

    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

  • 38 Pages midterm_2_practice
    Midterm_2_practice

    School: UCLA

    Course: Probability Theory

    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

  • 38 Pages midterm_2_practice-solutions
    Midterm_2_practice-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 35 Pages midterm_3
    Midterm_3

    School: UCLA

    Course: Probability Theory

    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

  • 41 Pages midterm_3_practice
    Midterm_3_practice

    School: UCLA

    Course: Probability Theory

    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

  • 38 Pages midterm_1_practice_1
    Midterm_1_practice_1

    School: UCLA

    Course: Probability Theory

    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

  • 38 Pages midterm_1
    Midterm_1

    School: UCLA

    Course: Probability Theory

    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

  • 28 Pages homework8
    Homework8

    School: UCLA

    Course: Probability Theory

    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

  • 1 Page problem_set_2
    Problem_set_2

    School: UCLA

    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

  • 2 Pages problem_set_3
    Problem_set_3

    School: UCLA

    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

  • 1 Page problem_set_4
    Problem_set_4

    School: UCLA

    Math 170B Section 1 Winter 2011 Problem Set # 4 Be sure to put your section number & TA name on the cover of your problem set. Problem (1) Show that for any random variable X, and any a > 0 P(|X| > a) in two ways: (i) Deduce it from Markov's inequality. (

  • 1 Page problem_set_6
    Problem_set_6

    School: UCLA

    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

  • 2 Pages problem_set_5
    Problem_set_5

    School: UCLA

    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

  • 1 Page problem_set_2
    Problem_set_2

    School: UCLA

    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

  • 2 Pages problem_set_3
    Problem_set_3

    School: UCLA

    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

  • 2 Pages problem_set_4
    Problem_set_4

    School: UCLA

    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

  • 19 Pages homework1
    Homework1

    School: UCLA

    Course: Probability Theory

    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

  • 19 Pages homework2
    Homework2

    School: UCLA

    Course: Probability Theory

    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

  • 4 Pages 170Bp10s04a
    170Bp10s04a

    School: UCLA

    Course: Solution

    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:

  • 4 Pages 170Bp10s05a
    170Bp10s05a

    School: UCLA

    Course: Solution

    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.

  • 4 Pages 170Bp10s06a
    170Bp10s06a

    School: UCLA

    Course: Solution

    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

  • 4 Pages 170Bp10s07a
    170Bp10s07a

    School: UCLA

    Course: Solution

    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

  • 3 Pages 170Bp10s08a
    170Bp10s08a

    School: UCLA

    Course: Solution

    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

  • 3 Pages 170Bp10s09a
    170Bp10s09a

    School: UCLA

    Course: Solution

    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

  • 2 Pages 170Bp10s10
    170Bp10s10

    School: UCLA

    Course: Solution

    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

  • 2 Pages 170Bp10sm1a
    170Bp10sm1a

    School: UCLA

    Course: Solution

    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

  • 5 Pages 170Bp10s03a
    170Bp10s03a

    School: UCLA

    Course: Solution

    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

  • 4 Pages 170Bp10s02a
    170Bp10s02a

    School: UCLA

    Course: Solution

    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

  • 4 Pages 170Bp10s01a
    170Bp10s01a

    School: UCLA

    Course: Solution

    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?

  • 35 Pages homework5-solutions
    Homework5-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 25 Pages homework6
    Homework6

    School: UCLA

    Course: Probability Theory

    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 ,

  • 39 Pages homework6-solutions
    Homework6-solutions

    School: UCLA

    Course: Probability Theory

    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 ,

  • 28 Pages homework7
    Homework7

    School: UCLA

    Course: Probability Theory

    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/

  • 44 Pages homework7-solutions
    Homework7-solutions

    School: UCLA

    Course: Probability Theory

    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/

  • 3 Pages homework5-solutions
    Homework5-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 3 Pages homework4-solutions
    Homework4-solutions

    School: UCLA

    Course: Probability Theory

    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

  • 4 Pages solution_4
    Solution_4

    School: UCLA

    Course: Solution

    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

  • 2 Pages 170Bp10sm2a
    170Bp10sm2a

    School: UCLA

    Course: Solution

    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

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