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##### MATH 170b - Probability Theory - UCLA Study Resources
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###### 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

• 32 Pages
###### 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

• 31 Pages
###### 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

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• ###### homework6-solutions winter
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###### 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 ,

• 3 Pages
###### 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

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###### 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

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###### 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

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

• 2 Pages
###### 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

• 4 Pages
###### 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

• 4 Pages
###### 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
###### 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
###### 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:

• 5 Pages
###### 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

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

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?

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###### 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

• 6 Pages
###### 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) =

• 2 Pages
###### 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

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###### 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

• 4 Pages
###### 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

• 1 Page
###### 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

• 2 Pages
###### 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)

• 2 Pages
###### 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

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###### 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

• 1 Page
###### 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
###### 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
###### 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
###### 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

• 9 Pages
###### 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

• 2 Pages
###### 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

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School: UCLA

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###### 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

• 3 Pages
###### 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

• 41 Pages
###### 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

• 38 Pages
• ###### midterm_1_practice_2-solutions
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###### 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) =

• 29 Pages
###### 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

• 44 Pages
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###### 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

• 38 Pages
###### 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

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

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

• 2 Pages
###### 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

• 2 Pages
###### 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

• 1 Page
###### 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

• 133 Pages
###### 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

• 2 Pages
###### 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_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

• 1 Page
###### 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. (

• 2 Pages
###### 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_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

• 35 Pages
###### 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

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

• 44 Pages
###### 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/

• 28 Pages
###### 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/

• 39 Pages
###### 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 ,

• 25 Pages
###### 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 ,

• 35 Pages
###### 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

• 33 Pages
###### 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

• 39 Pages
###### 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

• 28 Pages
###### 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

• 36 Pages
###### 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

• 19 Pages
###### 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

• 19 Pages
###### 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

• 41 Pages
###### 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_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

• 35 Pages
###### 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_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

• 38 Pages
• ###### midterm_2_practice-solutions
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###### 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

• 1 Page
###### 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

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