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MATH 170b  Probability Theory  UCLA Study Resources

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

Homework1solutions
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

Homework4solutions
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

Midterm_1_practice_2solutions
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) =

Midterm_3solutions
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

Midterm_2solutions
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

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

Homework3solutions
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

Homework6solutions 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/probsupp.html And also the problems below: Problem 1. Denote points P0 ,

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/probsupp

Homework2solutions
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

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

PracticeFinal_SolsSec1
School: UCLA

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 > tX = x). P(Y > t). Problem 2. Le

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

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

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

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

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

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

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

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

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 BerryEssen Theorem. Problems: (1) Let X1 , X2 , . . . be i.i.d and uniformly distribute

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

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)

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

BerryEsseen
School: UCLA
The BerryEsseen 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 BerryEsseen inequality is (x) p FSn (x) n 3 , (BE) whe

SolutionsHwk1
School: UCLA

SolutionsHwk2
School: UCLA

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

Homework 3 Solutions
School: UCLA
Course: Probability
Probability Theory, Math 170B, Spring 2013 Note: Although solutions exist online, 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

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/probsupp.html And also the problems

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

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

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/probsupp.htm

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

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

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

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[XY ]Z] = E[X]. Apply this formula to the following problem: Roll a far 6sided die and observe the number Z that came up. Then toss a fair coin

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

SolutionsHwk3
School: UCLA

SolutionsHwk6
School: UCLA

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

Homework1solutions
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 > tX = x). P(Y > t). Solution: 1.

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:

Homework2solutions
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

Midterm_2_practice_1solutions
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 nonnegative

Midterm_2_practice2
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

Midterm_2_practice2solutions
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

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/probsupp.html And also the pro

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/probsupp.html And also

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) =

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

Midterm_1_practice_1solutions
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

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

Midterm_1solutions
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

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

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

Midterm_2_practicesolutions
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

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

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

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

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

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

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

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

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. (

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

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

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

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

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

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

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

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:

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.

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

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

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

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

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

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

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

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) = xex(y+1) 1cfw_x,y>0 . Calculate (a) E eXY (1+Y )2 (b) E(X) xy (c) E(XY ) 1 x x = dxxe

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?

Homework5solutions
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/probsupp

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/probsupp.html And also the problems below: Problem 1. Denote points P0 ,

Homework6solutions
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/probsupp.html And also the problems below: Problem 1. Denote points P0 ,

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/

Homework7solutions
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/

Homework5solutions
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/probsupp

Homework4solutions
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

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

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