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Chapter 6 Expected Value and Variance 6.1 Expected Value of Discrete Random Variables When a large collection of numbers is assembled, as in a census, we are usually interested not in the individual numbers, but rather in certain descriptive quantities such as the average or the median. In general, the same is true for the probability distribution of a numerically-valued random variable. In this and in the next section, we shall discuss two such descriptive quantities: the expected value and the variance. Both of these quantities apply only to numerically-valued random variables, and so we assume, in these sections, that all random variables have numerical values. To give some intuitive justification for our definition, we consider the following game. Average Value A die is rolled. If an odd number turns up, we win an amount equal to this number; if an even number turns up, we lose an amount equal to this number. For example, if a two turns up we lose 2, and if a three comes up we win 3. We want to decide ifif a two turns up we lose 2, and if a three comes up we win 3.

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Berkeley - MATH - 54257

4.3. GENERALIZED PERMUTATIONS AND COMBINATIONS674.3. Generalized Permutations and Combinations 4.3.1. Permutations with Repeated Elements. Assume that we have an alphabet with k letters and we want to write all possible words containing n1 times the rst

Berkeley - MATH - 54257

Math 110, Extra problems GSI: Dario1. Let V, W, Z be nite dimensional vector spaces and f : V W , g : V Z linear maps. Prove that N (f ) is contained in N (g ) if and only if there exists a linear map L : W Z such that g = L f .2. Find all the matrices

Berkeley - MATH - 54257

Charley CrissmanMath 55 Discussion Notes Injectivity and SurjectivitySeptember 15th, 2010How to Prove Injectivity Suppose we want to show that the function h : A B is injective. The proof always begins the same way: Proof. Let a1 , a2 A be arbitrary, a

Berkeley - MATH - 54257

Charley CrissmanMath 55 Discussion Notes Permutations and CombinationsOctober 13th, 2010Question: Recall that a poker hand consists of 5 cards from a standard 52-card deck. How many ways are there to get a: 1. four-of-a-kind? 2. two-of-a-kind? 3. hand

Berkeley - MATH - 54257

A NOTE ON HYPERVECTOR SPACESarXiv:1002.3816v1 [math.GM] 19 Feb 2010SANJAY ROY , T. K. SAMANTADepartment of Mathematics, South Bantra Ramkrishna Institution , India. Department of Mathematics, Uluberia College,West Bengal, India. E-mail : sanjaypuremath

Berkeley - MATH - 54257

Section 3Prime Numbers and Prime FactorisationDefinition 3.1 A prime number is an integer p > 1 whose only positive divisors are 1 and p. Example 3.2 The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, . . . . See the Prime

Berkeley - MATH - 54257

Math 110 Midterm ExamProfessor K. A. Ribet September 26, 2002Please put away all books, calculators, digital toys, cell phones, pagers, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Please write your name on each

Berkeley - MATH - 54257

Chapter 1 Vector SpacesPer-Olof Perssonpersson@berkeley.eduDepartment of Mathematics University of California, BerkeleyMath 110 Linear AlgebraVector SpacesDenition A vector space V over a eld F is a set with the operations addition and scalar multip

Berkeley - MATH - 54257

2010-2011College of ChemistryGuideto Undergraduate Studies in Chemistry, Chemical Engineering, and Chemical BiologyUniversity of California, BerkeleyAcademic Calendar2010-11Fall Semester 2010Tele-BEARS Begins Fee Payment Due Fall Semester Begins W

Berkeley - MATH - 54257

PROCEDURES FOR REQUESTING CHANGE OF COLLEGE TO CHEMISTRY (FROM L&S, ENGINEERING, ETC.)Fill in the blanks below, attach the requested information, and turn this form in to Maura in the College of Chemistry Undergraduate Majors Office in 420 Latimer Hall.

Berkeley - MATH - 54257

Combinatorial Laplacians of Simplicial ComplexesA Senior Project submitted to The Division of Natural Science and Mathematics of Bard College by Timothy E. GoldbergAnnandale-on-Hudson, New York May, 2002AbstractIn this paper, we study the combinatoria

Berkeley - MATH - 54257

University of California, BerkeleySee reverse for instructions. Please type or print in ink.Undergraduate Petition Change of College, Major or CurriculumFor Term: Fall Spring Year:100019Note: To declare or change a major in the College of Letters and

Berkeley - MATH - 54257

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A02A BIJECTIVE PROOF OF fn+4 + f1 + 2f2 + + nfn = (n + 1)fn+2 + 3 Philip Matchett Wo o dDepartment of Mathematics, Rutgers University (New Brunswick), Hil l Center-Busch Campus, 110

Berkeley - MATH - 54257

4. LINEAR TRANSFORMATIONSDR. TAEIL YILinear Algebra with Applications 6th ed. -Steven J. LeonLECTURE NOTE12DR. TAEIL YI4.1 Denition and Examples Linear Transformation A mapping L from a vector space V into a vector space W is said to be a linear tra

Berkeley - MATH - 54257

Practice Problems (1) 1/18/06 A m n matrix A is called u pper triangular if all the entries lying below the diagonal entries are zero. That is, Aij = 0, whenever i > j . Prove that the upper triangular matrices form a subspace of Mmn (F ). Solution: Clear

Berkeley - MATH - 54257

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Berkeley - MATH - 54257

Introduction to Number Theory Notes 1. Divisibility in Z Denition 1.1 Let a and b are integers with a = 0. We say that a divides b if there is an integer c such that b = ac. Terminology: a is a divisor of b, a is a factor of b, b is a multiple of a Notati

Berkeley - MATH - 54257

Music 128T/TM/Theater 125: The American Musical Study Guide for Listening Quiz #1 After the Ball Charles K. Harris, 1892 The Bowery Gaunt and Hoyt, 1891 I Got Rhythm George Gershwin and Ira Gershwin, 1930 Smoke Gets in Your Eyes Kern and Harbach, 1933 (no

Berkeley - MATH - 54257

pp. 105 Exercise 2. Let B = cfw_1, 2, 3 be the basis for C3 dened by1 = (1, 0, 1) 2 = (1, 1, 1) 3 = (2, 2, 0). Find the dual basis of B . The rst element of the dual basis is the linear function 1 such that 1 (1 ) = 1, 1 (2 ) = 0 and 1 (3 ) = 0. To descr

CUHK - MEEM - 2029

MEEM 2029Course Aims & Objectives: To understand, operate and maintain electrical systems To be familiar with basic knowledge of electrical engineering To learn the basic calculation/analysis of electrical systems To communicate and cooperate with electr

UCLA - CHEM - 110A

UT Arlington - CSE - 2315

CSE 2315 - Discrete Structures Homework 3: Sets and CombinatoricsCSE 2315 - Discrete StructuresHomework 3- Fall 2010 Due Date: Oct. 28 2010, 3:30 pmSets1. Rewrite the following sets as a list of elements. a) cfw_x|(y )(y N x = y 3 x < 30) b) cfw_x|x i

UT Arlington - CSE - 2315

CSE 2315 - Discrete Structures Homework 1: Propositional Calculus and Predicate LogicCSE 2315 - Discrete StructuresHomework 1- Fall 2010 Due Date: Sept. 16 2010, 3:30 pmStatements, Truth Values, and Tautologies1. Which of the following are statements

UT Arlington - CSE - 2315

CSE 2315 - Discrete Structures Homework 1: Propositional Calculus and Predicate LogicCSE 2315 - Discrete StructuresHomework 1- Solutions - Fall 2010 Due Date: Sept. 16 2010, 3:30 pmStatements, Truth Values, and Tautologies1. a) Is a statement. b) Is a

UT Arlington - CSE - 2315

CSE 2315 - Discrete Structures Homework 2: Predicate Calculus and Proof TechniquesCSE 2315 - Discrete StructuresHomework 2- Fall 2010 Due Date: Oct. 7 2010, 3:30 pmProofs using Predicate LogicFor all your predicate logic proofs you can use only the ru

UT Arlington - CSE - 2315

CSE 2315 - Discrete Structures Homework 2: Predicate Calculus, Proof Techniques, and RecursionCSE 2315 - Discrete StructuresHomework 2- Solutions - Fall 2010 Due Date: Oct. 7 2010, 3:30 pm1. a) 1. (x)R(x) 3. (y )(R(y ) P (z, y, z ) 4. 5. b) R(z ) P (z,

Mid-America - ACCOUNTING - Act8484

CHAPTER 1FINANCIAL ACCOUNTING AND ACCOUNTING STANDARDSIFRS questions are available at the end of this chapter.TRUE-FALSE-ConceptualAnswerF T T T F F F F T T F F T T F T F T T FNo.1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20

UNSW - ECON - 2112

Final-Practice Exam #2Economics 101 Professor H. Quirmbach Final ExamPRINT NAME_ STUDENT ID NO._ GROUP TIME_SCORE_ INSTRUCTIONS: 1. Fill in all requested information above and on the answer sheet.2. There are 40 multiple choice questions and one probl

University of Massachusetts Boston - MATH - MA260

solutions for linear algebra course

University of Massachusetts Boston - MATH - MA260

Math 21b Solutions Week 1 Lecture 2 Section 1.2: 4,10,18,22,42,32*,20* TF6,44*

University of Massachusetts Boston - MATH - MA260

Math 21b Solutions: Week 1 Lecture 3 1.3: [12,14],28,36,56,62,26*,46* TF23,41*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 2 Lecture 4 Section 2.1: 6, [18,22], 24-30, 38, 44, 34*, 52* TF5,6*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 2 Lecture 5 Section 2.2: 6,28,30,34,38,47*,36* TF14,37*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 2 Lecture 6 Section 2.3: 16,30,[46,48],40*,42* Section 2.4: 14,[67-71],28*,72* TF26,49*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 3 Lecture 7 3.1: [10,22],34,42,44,54,38*,48* TF8,50*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 3 Lecture 8 3.2: 18,[30,32],40,48,54,36*,38* TF16,24*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 4 Lecture 9 3.3: 16,26,32,38,72,36*,84* TF2,6*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 4 Lecture 10 3.4: 10,16,42,50,64,32*,26* TF24,34*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 4 Lecture 11 4.1: 6-8,9-11,36,40,58,44*,12*-15* TF2,18*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 5 Lecture 12 4.1: 1-5,14,34,44,52,42*,56* TF18/30*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 5 Lecture 13 Section 5.1: 8,16,18,20,28,38*,14* TF5/14*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 6 Lecture 14 Section 5.2: 10,14,18,36,40,44*,38* TF29*/48*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 6 Lecture 15 Section 5.3: 5-8,9-11,13-16,17-20,44,48*,46* TF22*/23*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 6 Lecture 16 Section 5.4: 4,10,24,34,40,20*,18* TF24*,18*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 7 Lecture 17 Section 6.1: 40,42,48,54,56,45*,60* TF4*,30*

University of Massachusetts Boston - MATH - MA260

Math 21b Solutions Week 7 Lecture 18 Section 6.2: 4,10,40,66,50*,44* and 6.3: 12 TF32*,34*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 7 Lecture 19 Section 7.1: 10,36 and 7.2: 12,28,40,25*,26* TF6*,20*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 8 Lecture 20 Section 7.3: 14,18,20,28,44,32*,36* TF25*,51*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 8 Lecture 21 Section 7.4: 18,30,36,54,60,58*,56* TF23*,29*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 8 Lecture 22 7.5: 2(but for z8),14,26,32,38,50*,36* TF11*,41*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 9 Lecture 23 Section 7.6: 10,16,22,34,42, 38*,40*, TF*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 9 Lecture 24 Section 8.1: 4,10,24,28,42,36*,26* TF15*,16*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 10 Lecture 25 Section 9.1: 28,32,40,42,54,24*,46*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 10 Lecture 26 Section 9.2: 12,18,22-26,32,40,36* TF 19*

University of Massachusetts Boston - MATH - MA260

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 11 Lecture 28 Section 4.2: 32,54,50,66,78* Section 9.3: 4 TF44*58*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 11 Lecture 29 Section 9.3: 16,26,34,40,44,36*,20*

University of Massachusetts Boston - MATH - MA260

Math21b Solutions Week 11 Lecture 30 Section 5.5: 4,10,16,24,32,20*

University of Massachusetts Boston - MATH - MA260

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University of Massachusetts Boston - MATH - MA260

Math 2210: HW #1 Solutions February 3rd, 2010

University of Massachusetts Boston - MATH - MA260

MATHEMATICS 2270 Introduction to Linear Algebra Spring Semester 2010Time: TuesdayThursday, 4:35-6:15pm, JTB 130 Instructor: Professor Grant B. Gustafson1 , JWB 113, 801-581-6879. Oce Hours: JWB 113, TH 3:30-4:20pm. Other times will appear on my door card