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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
School: UCLA
Course: Math 170
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:
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
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:
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 3 solutions 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:/www.athenasc
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
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 6 Due Friday, May 8th From the books supplementary problems, solve problems 5, 6, 7, 9 and 19 in Chapter 7 (see http:/www.athenasc.com/prob-supp.html). And also the problems below: Pro
School: UCLA
Course: Math32A
Chapter Review Exercises SOLUTION 811 The function whose value we want to approximate is f (x, y, z) = x 2 + y2 + z We use the linear approximation at the point (7, 5, 70), hence h = 7.1 - 7 = 0.1, k = 4.9 - 5 = -0.1, and l = 70.1 - 70 = 0.1. We g
School: UCLA
Course: Math32A
MULTIVARIABLE CALCULUS (MATH 32A) FALL 2004 LECTURE NOTES 10/1/04 Office Hours: MS 7344, 11:00 12:40 Mon., Wed., Fri. E-mail: lchayes@math.ucla.edu Homework: 20% (each problem graded on a 2-pt. scale). PRINT OUT HOMEWORK FROM COURSE WEBPAGE Midterm
School: UCLA
Course: Linear Algebra
Wtd. Week 5 VHS Reed.“ ‘Frvm Mmdaj: Given ate? o‘c‘W-C—‘i‘ors m Rod ‘7; is mduvxdavﬁ'i‘c. H’SA llmr comb‘nac‘ﬁon 0F Tigﬁmy'ikﬂ 11m. Set- oF «chars Is ling.ka Wm H: and «F M Veehars are. rm Ome-wtse) ﬁn. Sul- 15 [WU-Art: \ndech '3 A hum nhhm among We. vect
School: UCLA
Course: Casualty Loss Models 2
Chapter 14 Simulation Simulation The objective in performing a simulation is to reproduce the behavior of a random variable by generating observations from another random variable which has the same distribution as the
School: UCLA
Course: SYSTMS-DIFFNTL EQTN
Math 33B Time table: Lecture time Lecture 2 Lecture 3 Lecture location Midterm 1 Midterm 2 Final time 1111:50am MWF 4000A MS October 31st , Friday. November 24th , Wednesday lecture time, lecture room lecture time, lecture room December 13, 11:30am2:30pm
School: UCLA
Course: MULTIVARIABLE CALCULUS
Mathematica for Rogawski's Calculus 2nd Edition 2010 Based on Mathematica Version 7 Abdul Hassen, Gary Itzkowitz, Hieu D. Nguyen, Jay Schiffman W. H. Freeman and Company New York 2 Mathematica for Rogawski's Calculus 2nd Editiion.nb Copyright 2010 Mathem
School: UCLA
Course: Actuarial Math
172C June 2013 AV, Asset Shares, Year by Year Financial Universal Life Account Values 1. For Universal Life UL policies in particular, the Account Value AV functions like a policy holder "bank account that is periodically updated Even though premium
School: UCLA
Course: Optimization
Lecture 17: The Dual Problem 1 Dual of Canonical Linear Programs For every linear programming problem (original, primal problem), there is a companion problem, called the dual linear program, where the roles of variables and constraints are reversed. That
School: UCLA
Course: Optimization
Lecture 15: Initialization of Simplex Method 1 The Two-Phase Method Phase-1: nd a basic feasible solution for the problem which has the original set of constraints and the objective minimize z = ai i where cfw_ai are the articial variables. If one artic
School: UCLA
Course: Optimization
Lecture 14: Degeneracy and Initialization 1 Multiple Solutions Example 1. Solve the following linear program using the simplex method minimize z= x1 subject to 2x1 + x2 2 x1 + x2 3 x1 3 x 1 , x2 0. Solution. We rst convert the problem into standard form b
School: UCLA
Course: Optimization
Lecture 13: Simplex Method Using Tableaus Advantages of tableaus: More organized than the formulas 1) Give all information (basic feasible solution, reduced cost, B in a table Change basis by row operations in the table Make it possible to solve small
School: UCLA
Course: Optimization
Lecture 12: Simplex Method II Consider the linear program in standard form minimize z = cT x subject to Ax = b x 0. 1. Optimality Test. Since BxB + NxN = b =) xB = B 1 b B 1 NxN , which is the general formula for xB , we have z = c T x = cT x B + c T x N
School: UCLA
Course: Optimization
Lecture 11: Simplex Method I Simplex Method General procedure to solve a linear program in standard form: minimize z = cT x subject to Ax = b, x 0. Find all basic feasible solutions (or equivalently extreme points of the feasible set). Choose an initial
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
School: UCLA
Course: ANALYSIS
Math 131a Lecture 2 Spring 2009 Midterm 1 Name: Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the b
School: UCLA
Course: Math 131a
Mathematics 131A Fall 2007 FINAL Your Name: Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Good luc
School: UCLA
Course: Math 131a
Mathematics 131A - Final Examination Instructor : D. E. Weisbart June 12, 2012 NAME (please print legibly): Your University ID Number: Signature: There are SEVEN questions on this examination. Calculators, notes and books may not be used in this examina
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
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
School: UCLA
Course: Math 170
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:
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
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:
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 3 solutions 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:/www.athenasc
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
School: UCLA
Math 172 B Class Project 1 Object: to download a specific mortality table from the Society of Actuaries website and utilize mortality rates to calculate life expectancies Steps: 1. Log onto www.soa.org 2. At the bottom of the opening page, select popular
School: UCLA
Course: Math
Math 31A Lecture 5 Summation Notation November 7, 2015 Summation notation is a short-hand way of writing sums of a lot of numbers when there is a pattern. For example, we could write 1 + 2 + 3 + + 100 as 100 j j=1 The symbol indicates that this is a sum.
School: UCLA
Course: Geometry A
Chapter4 TrianglesBasic Geometry InequalitiesinTriangles Anglesandtheiroppositesidesintrianglesarerelated.Infact,thisisoftenreflectedinthe labelingofanglesandsidesintriangleillustrations. Anglesandtheiroppositesidesareoften labeledwiththesameletter.Anuppe
School: UCLA
Course: Geometry A
Chapter4 TrianglesBasic Geometry CentersofTriangles Thefollowingareallpointswhichcanbeconsideredthecenterofatriangle. Centroid(Medians) Thecentroidistheintersectionofthethreemediansofatriangle.Amedianisa linesegmentdrawnfromavertextothemidpointofthelineop
School: UCLA
Course: Geometry A
Chapter3 ParallelandPerpendicularLines Geometry ProvingLinesareParallel Thepropertiesofparallellinescutbyatransversalcanbeusedtoprovetwolinesareparallel. CorrespondingAngles Iftwolinescutbyatransversalhavecongruentcorrespondingangles, thenthelinesareparal
School: UCLA
Course: Geometry A
Chapter4 TrianglesBasic Geometry TypesofTriangles Scalene AScaleneTrianglehas3sidesofdifferent lengths.Becausethesidesareof differentlengths,theanglesmustalsobe ofdifferentmeasures. Equilateral AnEquilateralTrianglehasall3sidesthe samelength(i.e.,c
School: UCLA
Course: Geometry A
Chapter3 ParallelandPerpendicularLines Geometry ParallelLinesandTransversals Transversal B A C Consecutive:referstoanglesthatare onthesamesideofthetransversal. D Alternate:referstoanglesthatareon oppositesidesofthetransversal. ParallelLines F E G Interior
School: UCLA
Course: Introduction Of Complex Analysis
Math 132: Complex Analysis for Applications INSTRUCTOR: Kefeng Liu TIME and LOCATION: MWF 9:00-9:50AM, MS5117. OFFICE HOURS: TT 9:00-10:00AM in MS 6330 (Or by appointment) SYLLABUS: Official Syllabus Text: Complex Analysis by Tom Gamelin TA: Wenjian Liu,
School: UCLA
Course: Linear Algebra
Math 33A, Sec. 2 Linear Algebra and Applications Spring 2012 Instructor: Professor Alan J. Laub Dept. of Mathematics / Electrical Engineering Math Sciences 7945 phone: 310-825-4245 e-mail: laub@ucla.edu Lecture: MWF 2:00 p.m. 2:50 p.m.; Rolfe 1200 Office
School: UCLA
Course: Math
Dierential and Integral Calculus Math 31A, Lecture 5 Fall Quarter 2015 MWF 9:00am-9:50am, MS 6221 Instructor: Melissa Lynn mklynn@math.ucla.edu MS 3915B My Oce Hours: Mondays 11:30am-12:30pm, 3:30pm-4:30pm Fridays 10am-11am or by appointment Teaching Assi
School: UCLA
Course: CALCULUS OF A SINGLE VARIABLE
MATH 31A, Dierential and Integral Calculus, Lecture 4, Fall 2015 Exterior Course Website: http:/www.math.ucla.edu/heilman/31af15.html Prerequisite: Successful completion of Mathematics Diagnostic Test or course 1 with a grade of C- or better. Course Conte
School: UCLA
Course: Algebra-applicatns
v P y V ` V ` x G ~ y | G V | ` ` F G G y V F F GI | V U )HoASY2dddI)dY)2vdG u !A)w u zAdA)YV y V ` V ` x G ~ | G V | ` ` G V F GI X y V G ` ` FI ` ` 6 8 4 8 1 HAa92ddd!I dY)2vdG u F AHWdYaHa2 9!4 20 P U b U y b q b Uv b pv b gv b i tAACttAAmYACt7t
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
School: UCLA
Course: Math 170
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:
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
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:
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 3 solutions 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:/www.athenasc
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
School: UCLA
Course: Actuarial Math
Sule Ozler Economics 12 Fall 2011 READ THE EXPLANATIONS IN BOLD LETTERS CAREFULLY! MID-TERM EXAM - GOOD LUCK! WRITE YOUR NAME AND ID# You will not be allowed to ask questions during the exam. This exam has three sections. Each section has multiple questio
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
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
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
School: UCLA
Course: Mathemtcl Modeling
Math 142-2, Homework 1 Solutions April 7, 2014 Problem 34.4 Suppose the growth rate of a certain species is not constant, but depends in a known way on the temperature of its environment. If the temperature is known as a function of time, derive an expres
School: UCLA
Course: Probability Theory
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
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
School: UCLA
Course: ANALYSIS
Math 131a Lecture 2 Spring 2009 Midterm 1 Name: Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the b
School: UCLA
Course: Math 131a
Mathematics 131A Fall 2007 FINAL Your Name: Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Good luc
School: UCLA
Course: PRECALCULUS PART 1
Math 115 Spring 11 Written Homework 10 Solutions 1. For following limits, state what indeterminate form the limits are in and evaluate the limits. 3x2 4x 4 x2 2x2 8 (a) lim 0 . Algebraically, we hope to be able to factor the 0 numerator and denominator an
School: UCLA
Course: Math 131a
Mathematics 131A - Final Examination Instructor : D. E. Weisbart June 12, 2012 NAME (please print legibly): Your University ID Number: Signature: There are SEVEN questions on this examination. Calculators, notes and books may not be used in this examina
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 ,
School: UCLA
Course: Probability Theory
Probability Theory, Math 170A, Fall 2014 - Homework 5 solutions From the textbook solve the problems 32, 39, 40 at the end of the Chapter 2. Solution to Problem 32: Let Xi be the indicator of the event that the rst person in the i-th couple is alive and Y
School: UCLA
Course: Probability Theory
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
School: UCLA
Course: Probability Theory
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
School: UCLA
Course: Linear Algebra And Applications
Name: Student ID: Prof. Alan J. Laub Section: 2 May 4, 2012 Math 33A/2 MIDTERM EXAMINATION Spring 2012 Instructions: (a) The exam is closed-book (except for one page of notes) and will last 50 minutes. No calculators, cell phones, or other electronic devi
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
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 ,
School: UCLA
Course: Actuarial Math
Math 172 B Class Project 1 Object: to download mortality tables from the Society of Actuaries website and utilize mortality rates for a selected table to calculate life expectancies Steps: 1. Log onto www.soa.org 2. Put the words Table Manager in the Sear
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 6 Due Friday, May 8th From the books supplementary problems, solve problems 5, 6, 7, 9 and 19 in Chapter 7 (see http:/www.athenasc.com/prob-supp.html). And also the problems below: Pro
School: UCLA
Course: Introduction To Discrete Structures
MATH 61: HOMEWORK 2 3.3 25. not reexive, not symmetric, antisymmetric, transitive, not a partial order 26. reexive, not symmetric, antisymmetric, transitive, partial order 28. reexive, symmetric, not antisymmetric, transitive, not a partial order (this is
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
School: UCLA
Course: Introduction Of Complex Analysis
Solutions for assignment 2 1 of 2 Section 2.2 Question 9: (a) (z-5i)^2 is a polynomial, hence continuous everywhere, and thus on e can simply substitute in the limit to obtain (2+3i)^2 - 5i)^2 = -8i. (c) This is an indeterminate form; one can either use L
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 6 Due Friday, May 8th From the books supplementary problems, solve problems 5, 6, 7, 9 and 19 in Chapter 7 (see http:/www.athenasc.com/prob-supp.html). And also the problems below: Pro
School: UCLA
Course: Math32A
Chapter Review Exercises SOLUTION 811 The function whose value we want to approximate is f (x, y, z) = x 2 + y2 + z We use the linear approximation at the point (7, 5, 70), hence h = 7.1 - 7 = 0.1, k = 4.9 - 5 = -0.1, and l = 70.1 - 70 = 0.1. We g
School: UCLA
Course: Math32A
MULTIVARIABLE CALCULUS (MATH 32A) FALL 2004 LECTURE NOTES 10/1/04 Office Hours: MS 7344, 11:00 12:40 Mon., Wed., Fri. E-mail: lchayes@math.ucla.edu Homework: 20% (each problem graded on a 2-pt. scale). PRINT OUT HOMEWORK FROM COURSE WEBPAGE Midterm
School: UCLA
Course: Linear Algebra
Wtd. Week 5 VHS Reed.“ ‘Frvm Mmdaj: Given ate? o‘c‘W-C—‘i‘ors m Rod ‘7; is mduvxdavﬁ'i‘c. H’SA llmr comb‘nac‘ﬁon 0F Tigﬁmy'ikﬂ 11m. Set- oF «chars Is ling.ka Wm H: and «F M Veehars are. rm Ome-wtse) ﬁn. Sul- 15 [WU-Art: \ndech '3 A hum nhhm among We. vect
School: UCLA
Course: Casualty Loss Models 2
Chapter 14 Simulation Simulation The objective in performing a simulation is to reproduce the behavior of a random variable by generating observations from another random variable which has the same distribution as the
School: UCLA
Course: Casualty Loss Models 2
Chapter 15 Simulation The Bootstrap Method And Statistical Analysis Using Simulation In statistics, bootstrapping can refer to any test or metric that relies on random sampling with replacement . Bootstrapping allows ass
School: UCLA
Course: Casualty Loss Models 2
Chapter 13 Credibility Theory Empirical Bayes Credibility Methods Non-parametric Empirical Bayes Credibility In the Bayesian and Buhlmann credibility approaches that have been considered, it was assumed that the conditiona
School: UCLA
Course: Casualty Loss Models 2
Chapter(12( Credibility(Theory( Buhlmann(Credibility( The(Buhlmann(Credibility(Structure( The"Buhlmann"model"has"the"same"initial"structure"as"the"Bayesian"credibility"model."There"is"a" population"or"portfolio"of"risks."Each"member"of"the"population"has"
School: UCLA
Course: Casualty Loss Models 2
Chapter(10( Credibility(Theory( Bayesian(Estimation,(Discrete(Prior( The"Bayesian"approach"to"credibility"has"the"same"initial"steps"as"the"Bayesian"analysis"presented"in" Chapter"9,"we"just"take"the"analysis"a"few"steps"further." We"begin"with"the"basic"
School: UCLA
Course: Casualty Loss Models 2
Chapter(11( Credibility(Theory( Bayesian(Estimation,(Continuous(Prior( The"application"of"Bayesian"analysis"to"credibility"estimation"can"be"summarized"as"follows;"The"random" variable"(usually"loss"frequency"or"severity,"or"perhaps"a"compound"aggregate"c
School: UCLA
Course: Casualty Loss Models 2
Chapter(9( Credibility(Theory( Bayesian(Estimation,(Discrete(Prior( History( According"to"WIKI(Thomas(Bayes"("c."1701"7"April"1761)"was"an"English"statistician,"philosopher"and" Presbyterian"minister,"known"for"having"formulated"a"specific"case"of"the"the
School: UCLA
Course: Casualty Loss Models 2
Chapter(8( Credibility(Theory( Limited(Fluctuation(Credibility( Introduction(to(Credibility(Theory( The"essential"objective"of"credibility"theory"is"to"estimate"the"mean"of"a"random"variable"!"from"a" random"sample"! , ! , , ! ." Credibility"estimation"ca
School: UCLA
Course: Casualty Loss Models 2
Chapter(7( Model(Estimation( Hypothesis(Testing(for(Fitted(Models( " Review(of(Hypothesis(Testing( Suppose"that"a"test"is"devised"to"try"to"determine"whether"or"not"a"coin"is"fair"(fair"means"that"there"is" 0.5"probability"of"heads"and"0.5"probability"of"
School: UCLA
Course: Casualty Loss Models 2
Chapter 6 Model Estimation Hypothesis Testing Review We can make a statement about a population parameter say the mean height of the population is equal to 510, collect a sample from that population, measure th
School: UCLA
Course: Casualty Loss Models 2
Chapter 3 Model Estimation Maximum Likelihood Estimation For Frequently Used Distributions Inverse Exponential for The inverse exponential distribution has density function = ! ! ! For the random sample ! , ! , , !
School: UCLA
Course: Casualty Loss Models 2
Chapter 5 Model Estimation Graphical Methods for Evaluating Estimated Models To help assess how well the estimated model fits to the original data, the Loss Models book describes several graphical comparisons of an
School: UCLA
Course: Casualty Loss Models 2
Chapter 2 Model Estimation Maximum Likelihood Estimation General Definition of Maximum Likelihood Estimation Maximum likelihood estimation is a method that is applied to estimate the parameters in a parametric distri
School: UCLA
Course: Casualty Loss Models 2
Chapter 1 Model Estimation Parametric Estimation, Method of Moments and Percentile Matching What is a Model? A model is an imitation of a real world system or process. Models of many activities can be developed,
School: UCLA
Course: Multivariable Calculus
Week 3: Polar, Cylindrical, Spherical Coordinates Integration in Polar Coordinates For some regions R it is convenient to use polar coordinates when evaluating double integrals R f (x, y)dA. These regions may be neither horizontally nor vertically simple
School: UCLA
Course: Multivariable Calculus
Ma 32B Week 2: Double and Iterated Integrals Double Integrals Over More General Regions Last week we dened the double integral f dA [a,b][c,d] over a rectangle [a, b] [c, d] in R2 . Recall that this quantity can be interpreted as the (signed) volume of th
School: UCLA
Course: Multivariable Calculus
Ma 32B Week 1 Notes Interpretations of the Double Integral Let R be a rectangle in the plane and f : R R a continuous function on R. One can interpret the double integral f (x, y)dA R in several ways. Volume under a surface: if f is nonnegative, then R f
School: UCLA
Course: Math
Izﬁg *2 V V“ = [26. m ‘7 {76tﬁr‘va ME} «3:. 5,5» Cm f f , to NM «9 m2 gin-Wm, E»; v ‘35: {aka QM? m- ay 5'7 & ea “ﬂab/“(’6 V“; M (“ii p! { 0AM}: S, W 044 5A mg”? 0a ’6’ *6. WM? .4" W . wick/{7&3 ‘ ’< (I Fremwm M Végervw (he. Pvtem‘emm : Ngng F \d/{M Cam
School: UCLA
Course: Math
E(€~ WCYQMW )1ch CDF§CKO€€> WWW~WMMAWM”WVAM CVQM” WWDéé/ég Lgﬂaﬁﬂﬁévpg: 35% ’POC{(wye, 94m“ M gwu ~ Pu? giécvbméf’ Meow/A .gyxfmwcx CW,9W,?“§“ eﬂcfpoeﬂﬁbaa a,” “$1; SWM a? ’gftl‘jvl/Z. J/V‘ 0 MWVLW Q’meéwvv > 06 JA'CC/TLQPMJE‘ H MOM 1 Diva/[’5‘ 1CD. 44
School: UCLA
Course: Math
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School: UCLA
Course: Math
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School: UCLA
Course: SYSTMS-DIFFNTL EQTN
Math 33B Time table: Lecture time Lecture 2 Lecture 3 Lecture location Midterm 1 Midterm 2 Final time 1111:50am MWF 4000A MS October 31st , Friday. November 24th , Wednesday lecture time, lecture room lecture time, lecture room December 13, 11:30am2:30pm
School: UCLA
Course: MULTIVARIABLE CALCULUS
Mathematica for Rogawski's Calculus 2nd Edition 2010 Based on Mathematica Version 7 Abdul Hassen, Gary Itzkowitz, Hieu D. Nguyen, Jay Schiffman W. H. Freeman and Company New York 2 Mathematica for Rogawski's Calculus 2nd Editiion.nb Copyright 2010 Mathem
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Course: Actuarial Math
172C June 2013 AV, Asset Shares, Year by Year Financial Universal Life Account Values 1. For Universal Life UL policies in particular, the Account Value AV functions like a policy holder "bank account that is periodically updated Even though premium
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Course: Optimization
Lecture 17: The Dual Problem 1 Dual of Canonical Linear Programs For every linear programming problem (original, primal problem), there is a companion problem, called the dual linear program, where the roles of variables and constraints are reversed. That
School: UCLA
Course: Optimization
Lecture 15: Initialization of Simplex Method 1 The Two-Phase Method Phase-1: nd a basic feasible solution for the problem which has the original set of constraints and the objective minimize z = ai i where cfw_ai are the articial variables. If one artic
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Course: Optimization
Lecture 14: Degeneracy and Initialization 1 Multiple Solutions Example 1. Solve the following linear program using the simplex method minimize z= x1 subject to 2x1 + x2 2 x1 + x2 3 x1 3 x 1 , x2 0. Solution. We rst convert the problem into standard form b
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Course: Optimization
Lecture 13: Simplex Method Using Tableaus Advantages of tableaus: More organized than the formulas 1) Give all information (basic feasible solution, reduced cost, B in a table Change basis by row operations in the table Make it possible to solve small
School: UCLA
Course: Optimization
Lecture 12: Simplex Method II Consider the linear program in standard form minimize z = cT x subject to Ax = b x 0. 1. Optimality Test. Since BxB + NxN = b =) xB = B 1 b B 1 NxN , which is the general formula for xB , we have z = c T x = cT x B + c T x N
School: UCLA
Course: Optimization
Lecture 11: Simplex Method I Simplex Method General procedure to solve a linear program in standard form: minimize z = cT x subject to Ax = b, x 0. Find all basic feasible solutions (or equivalently extreme points of the feasible set). Choose an initial
School: UCLA
Course: Optimization
Lecture 10: Representation of Solutions and Optimality Proposition 1. Consider S = cfw_Ax = b, x 0 for the linear program in standard form. If d = 0 is a direction of unboundedness of S, then Ad = 0 and d 0. Proof. Let x be a feasible point for the linear
School: UCLA
Course: Optimization
Lecture 9: Extreme Point and Direction of Unboundedness 1 Properties of Extreme Points Recall that x in a convex set S is an extreme point if there do not exist y, z S (y, z = x) and (0, 1) such that x = y + (1 )z. Theorem 1. Consider the linear program
School: UCLA
Course: Optimization
Lecture 8: Basic Solutions 1 Basic Solutions Denition 1. Consider a linear programming problem in standard form minimize z = cT x subject to Ax = b . x 0 Here x 2 Rn and A 2 Rmn with m n. We assume that A has full rank, i.e., rank(A) = m which implies tha
School: UCLA
Course: Optimization
Lecture 6: Convex Problems and Convex Functions 1 Global Minimizer of Convex Problems Theorem 1. If x is a local minimizer of a convex optimization problem, then x is a global minimizer. Moreover, if the objective function f is strictly convex, then x is
School: UCLA
Course: Optimization
Lecture 7: Concavity and Standard Form of Linear Programs 1 Concavity Given a convex set S, f (x) is convex on S if and only if f (x) is concave on S. Based on this property, we can decide the concavity of f by deciding the convexity of f . Example 1. f (
School: UCLA
Course: Optimization
Ming Yan i Math 164: Lecture #3 Notes June 30, 2014 Example: i The following gure illustrates the feasible region dened by the constraints i i g Optimality and Convexity Lecture 5:1 (x) = x1 + 2x2 + 3x3 6 = 0 g2 (x) = x1 0 Goal: book 2008/10/23 page 45 i
School: UCLA
Course: Optimization
Lecture 4: SVM and Feasibility 1 Support Vector Machines (SVM) Given a set of parameters about one subject (features) and a set of training points (the points with known labels), Support Vector Machines (SVM) can be used to classify 2 sets of data with di
School: UCLA
Course: Optimization
x 0 | 2 1 x1 Lecture 3: Nonlinear Program 1 1 Nonlinear Program Example 1. maximize f (x) = (x1 + x2 )2 subject to x1 x2 0 2x 1 Figure 1.3. Linear optimization problem.1 The feasible region is shaded. 2 x2 1. The feasible set is the shaded region shown
School: UCLA
Course: Optimization
Lecture 1: Optimization Models Goal: Mathematical modeling. Standard formulation of optimization problems. Feasible set. 1 Optimization The general procedure to solve a practical problem Problem Mathematical Modeling variables objective constraints Alg
School: UCLA
Course: Optimization
Lecture 2: Linear Programs Overview Denition 1. A linear program (optimization) involves the optimization of a linear function subject to linear constraints on the variables. Example 1. Suppose that a manufacturer of kitchen cabinets is trying to maximize
School: UCLA
Course: Probability Theory
i Probability: Theory and Examples Rick Durrett Edition 4.1, April 21, 2013 Typos corrected, three new sections in Chapter 8. Copyright 2013, All rights reserved. 4th edition published by Cambridge University Press in 2010 ii Contents 1 Measure Theory 1.1
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Course: MATH 31B
MATH 31B-3, Integration and Innite Series 2. Formula Sheet sec x dx = ln | tan x + sec x| + C If y = k(y b), then y = b + Cekt . Trig Formulas: 1. sin2 x + cos2 x = 1 2. tan2 x + 1 = sec2 x 1 cos 2x 3. sin2 x = 2 1 + cos 2x 4. cos2 x = 2 5. sin 2x = 2 sin
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Course: Analysis
Math 131A: lim sup and lim inf, subsequential limits, the ratio and root tests 1 lim sup and lim inf Suppose (sn ) is a bounded sequence. Notice that letting XN = cfw_sn : n > N we have cfw_sn : n N = X0 X1 X2 . . . and so, by homework 3, question 5)f )
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Course: Analysis
Math 131A: Series 1 Denitions and basic results n k=m ak Notation. Given a sequence (an ), we write for am + am+1 + . . . + an1 + an . n k=m ak Denition. Given a sequence (an ), the sequence (sn ) with sn = n=m of partial sums. If limn sn exists then we d
School: UCLA
Course: Analysis
Math 131A: sup, inf, countability 1 Suprema and Inma Denition. Let S be a nonempty subset of R. (a) If S is bounded above then the supremum of S is the number mincfw_M R : M is an upper bound for S, the least upper bound of S. We write sup S for it. (b) I
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Course: Analysis
Math 131A: monotone sequences, subsequences 1 Monotone sequences Once we prove that Cauchy sequences converge we are in good shape: we can then prove a sequence converges without knowing its limit. Lets deal with another situation where we can do the same
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Course: Analysis
Math 131A: Z, Q, algebraic numbers, rational zeros theorem, rings and elds 1 Z, Q, algebraic numbers We saw previously that the natural numbers are characterized by A notion of the next element. Ability to get a complete list starting from 1 and taking
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Course: Analysis
Math 131A: ordered elds, distance, the completeness axiom 1 Ordered elds and distance The rational numbers Q have additional structure. They have an order structure satisfying the following properties. Given a, b either a b or b a (totality) If a b and
School: UCLA
Course: Analysis
Math 131A: Sequences, Divergence, Algebra of limits 1 The denition of a divergent sequence Denition. If a sequence of real numbers (sn ) does not converge to any real number s, we say that it diverges. To understand exactly what it means for a sequence to
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Course: Analysis
Math 131A: diverging to , what is R? back to N 1 Diverging to In the second pdf, we disallowed the phrase tend to . This is a useful concept, however, and we wish to dene what we mean by it, although we will use sightly dierent language. Denition. A sequ
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Course: Analysis
Math 131A: Sequences, Convergence Black (1) then (1) white are (2) all I see (3) in my in-fan-cy, (5) red and yel-low then came to be (8). . . Tool, Los Angeles, 2001. 1 The word obvious As mentioned in the previous pdf, this course will deal with lots o
School: UCLA
Course: Probability Theory
Probability 170A Winter 2015 information sheet January 1, 2015 Instructor: Email: Eviatar B. Procaccia. procaccia@math.ucla.edu. Website: Textbook: https:/sites.google.com/site/ebprocaccia/teaching/math-170a-winter-2015. Introduction to Probability by D.
School: UCLA
Course: CALCULUS FOR LIFE SCIENCE STUDENTS
W. Conley Math 3a, Lecture 1 Fri, Nov 16, 2012 Midterm 2 Last Name: First Name: Student ID: Signature: TA: Section: David Taylor Athipat Thamrongthanyalak Tuesday Zhixin Zhou Thursday Instructions: Do not open this exam until instructed to do so. You will
School: UCLA
Course: CALCULUS FOR LIFE SCIENCE STUDENTS
W. Conley Math 3a, Lecture 1 Fri, Oct 26, 2012 Midterm 1 Last Name: First Name: Student ID: Signature: TA: Section: David Taylor Athipat Thamrongthanyalak Tuesday Zhixin Zhou Thursday Instructions: Do not open this exam until instructed to do so. You will
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
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Course: ANALYSIS
Math 131a Lecture 2 Spring 2009 Midterm 1 Name: Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the b
School: UCLA
Course: Math 131a
Mathematics 131A Fall 2007 FINAL Your Name: Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Good luc
School: UCLA
Course: Math 131a
Mathematics 131A - Final Examination Instructor : D. E. Weisbart June 12, 2012 NAME (please print legibly): Your University ID Number: Signature: There are SEVEN questions on this examination. Calculators, notes and books may not be used in this examina
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
School: UCLA
Course: Probability For Life Sciences Students
Math 3C Exam I Spring 2014 Name _ UCLA ID _ Lecture and Section _ No calculator, no cell phone, no notes. Show all of your work to receive credit . PROBLEM POINTS You earned : 1 5 2 5 3 5 4 5 5 10 6 10 7 10 Total 50 Simplify all answers Part I Counting :
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Course: Linear Algebra
Linear Algebra Math 115AH Midterm 1 Solutions Dominique Abdi 1. If W1 and W2 are subspaces of V and dim(W1 W2 ) = dim W1 what can you say about the relation between W1 and W2 ? Prove your answer. Solution. By the dimension theorem, dim(W1 + W2 ) = dim W1
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
School: UCLA
Course: Introduction Of Complex Analysis
25 Solution. Consider r1 and r2 such that 0 < R1 < r1 < r2 < R2 . Then f (z) is holomorphic on the (closed) annulus cfw_z : r1 |z| r2 . Thus by Cauchy Integral Theorem, we have f (z) = 1 2i 2 1 f () d z 2i 1 f () d, z where 1 = cfw_z : |z| = r1 and 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(). Solution: P() = 1 (b) (2 points) If P(A) =
School: UCLA
Course: Discrete Math
Practice First Test Mathematics 61 Disclaimer: Listed here is a selection of the many possible sorts of problems. actual test is apt to make a different selection. 1. Let S5 be the set of all binary strings of length 5 (such as 11001). The Let E be
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
School: UCLA
Course: Actuarial Math
Math 172A Exam 1 10/22/10 Name: _ 1. Deposit to fund: $3,000 paid on 1/1/2010 Withdrawal from fund: $1,000 on 1/1/2015 Interest accumulates at the following rates for the various periods specified: 2010 through 2012: 2013 through 2016: 2017 through 2022:
School: UCLA
Course: Actuarial Math
MATH 115A - Practice Final Exam Paul Skoufranis November 19, 2011 Instructions: This is a practice exam for the nal examination of MATH 115A that would be similar to the nal examination I would give if I were teaching the course. This test may or may not
School: UCLA
Course: Probability For Life Sciences Students
Math 3C Midterm 1 Solutions Name: BruinID: Section: Read the following information before starting the exam. This test has six questions and seven pages. It is your responsibility to ensure that you have all of the pages! Show all work, clearly and in o
School: UCLA
Course: Probability For Life Sciences Students
Math 3C Midterm 2 Name: BruinID: Section: Read the following information before starting the exam. This test has six questions and fourteen pages. It is your responsibility to ensure that you have all of the pages! Show all work, clearly and in order, i
School: UCLA
Course: CALCULUS OF A SINGLE VARIABLE
Math 31A Midterm 1 Solutions 1. [3 points] The derivative of (a) 1 (b) (c) x2 x)2 x3 Brent Nelson is: 1 x2 1 2 x3 (3x 1 1 + x2 + 8x2 x + 3) (d) None of the above. Solution. We rst apply the quotient rule d dx x2 x)2 x3 = d d x3 dx [(x2 x)2 ] (x2 x)2 dx (
School: UCLA
Course: Linear Algebra
Name: Student ID: Prof. Alan J. Laub June 9, 2011 Math 33A/1 FINAL EXAMINATION Spring 2011 Instructions: (a) The exam is closed-book (except for one two-sided page of notes) and will last two and a half hours. No calculators, cell phones, or other electro
School: UCLA
Course: Math32A
Math 32A: Practice Problems for Exam 2 Problem 1. A particle travels along a path r(t) with acceleration vector t4 , 4 t . Find r(t) if at t = 0, the particle is located at the origin and has initial velocity v0 = 2, 3 . Solution: We have r (t) = 1 1 t4 ,
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Course: Linear Algebra And Applications
MATH 33A/2 PRACTICE MIDTERM 2 Problem 1. (True/False) . 1 1 1 Problem 2. Let v1 = 1 , v2 = 0 , v3 = 1 . 0 1 1 3 (a) Show that B = 1 , v2 , v3 ) is a basis for R . (v 1 (b) Let u = 1 . Find the coordinates of u with respect to the basis B. 2 (c) Find the c
School: UCLA
Course: Math32A
Management 1A Winter 2004 Danny S. Litt Exam 3 Solutions Name: _ PROBLEM POINTS SCORE 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 TOTAL 100 Management 1A Problem 1 Winter 2004 (a) A company purchased a patent on January 1, 2002, for $1,000,
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Course: Math32A
MATH 32A FINAL EXAMINATION March 21th, 2012 Please show your work. You will receive little or no credit for a correct answer to a problem which is not accompanied by sucient explanations. If you have a question about any particular problem, please raise y
School: UCLA
Course: Math32A
Management 1A Summer 2004 Danny S. Litt EXAM 2 SOLUTION Name: _ Student ID No. _ I agree to have my grade posted by Student ID Number. _ (Signature) PROBLEM 1 2 3 4 5 6 7 8 TOTAL POINTS 20 25 20 20 30 30 30 25 200 SCORE MANAGEMENT 1A Problem 1
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Course: Math32A
MATH 32A FIRST MIDTERM EXAMINATION October, 18th, 20101 Please show your work. You will receive little or no credit for a correct answer to a problem which is not accompanied by sufficient explanations. If you have a question about any particular problem,
School: UCLA
Course: Probability Theory
Mathematics Department, UCLA P. Caputo Fall 08, midterm 1 Oct 22, 2008 Midterm 1: Math 170A Probability, Sec. 2 Last name First and Middle Names Signature UCLA ID number (if you are an extension student, say so) Please note: 1. Provide the information ask
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
School: UCLA
Course: Math 170
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:
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
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:
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 3 solutions 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:/www.athenasc
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
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
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
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Course: Mathemtcl Modeling
Math 142-2, Homework 1 Solutions April 7, 2014 Problem 34.4 Suppose the growth rate of a certain species is not constant, but depends in a known way on the temperature of its environment. If the temperature is known as a function of time, derive an expres
School: UCLA
Course: Probability Theory
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
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
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 ,
School: UCLA
Course: Probability Theory
Probability Theory, Math 170A, Fall 2014 - Homework 5 solutions From the textbook solve the problems 32, 39, 40 at the end of the Chapter 2. Solution to Problem 32: Let Xi be the indicator of the event that the rst person in the i-th couple is alive and Y
School: UCLA
Course: Probability Theory
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
School: UCLA
Course: Probability Theory
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
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 ,
School: UCLA
Course: Introduction To Discrete Structures
MATH 61: HOMEWORK 2 3.3 25. not reexive, not symmetric, antisymmetric, transitive, not a partial order 26. reexive, not symmetric, antisymmetric, transitive, partial order 28. reexive, symmetric, not antisymmetric, transitive, not a partial order (this is
School: UCLA
Course: Introduction Of Complex Analysis
Solutions for assignment 2 1 of 2 Section 2.2 Question 9: (a) (z-5i)^2 is a polynomial, hence continuous everywhere, and thus on e can simply substitute in the limit to obtain (2+3i)^2 - 5i)^2 = -8i. (c) This is an indeterminate form; one can either use L
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 ,
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 5 Due Friday, May 1st From the books supplementary problems, solve problems 21, 22, 27 and 28 in Chapter 4, as well as 1, 3 and 4 in Chapter 7 (see http:/www.athenasc.com/probsupp.html
School: UCLA
Course: Introduction Of Complex Analysis
EE101 Engineering Electromagnetics Winter 2013 1/8/13 Homework #1 Due: Thursday Jan 17 10:00 AM Hand in to the TA at beginning of class. No late homework is accepted (see grading policy posted on eeweb). If you cannot make it to class, you must slip the H
School: UCLA
Course: Math 170
Probability Theory, Math 170a, Winter 2015 - Homework 1 From the textbook solve the problems 2, 5-10 at the end of the Chapter 1. And also the problems below: Problem 1. Show that for any sets A and B P(A B) P(A) P(A B). Problem 2. We have a very weird di
School: UCLA
Course: Introduction Of Complex Analysis
132 Homework 1 solutions Exercise 1 Suppose |z| = 1 and |a| < 1. Show that, za =1 1 az Solution. First, we must show that the left hand side is even dened, i.e. we are not dividing by zero. To show that 1 az = 0 is the same as to show that 1 = az. As |az|
School: UCLA
Course: Math 170
Probability Theory, Math 170b, Spring 2015, Toni Antunovi c c Homework 6, solutions Due Friday, May 8th From the books supplementary problems, solve problems 5, 6, 7, 9 and 19 in Chapter 7 (see http:/www.athenasc.com/prob-supp.html). And also the problems
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
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
School: UCLA
Course: Introduction Of Complex Analysis
EE101 Engineering Electromagnetics Winter 2013 2/14/13 Homework #5 Due: Thursday Feb 21 10:00 AM Hand in to the TA at beginning of class. No late homework is accepted (see grading policy posted on eeweb). If you cannot make it to class, you must slip the
School: UCLA
Course: Math 170
Probability Theory, Math 170A - Homework 4 From the textbook solve the problems 16, 22, 24 at the end of the Chapter 2. Solve the problems 5 and 13 from the Chapter 2 additional exercises at http:/www.athenasc.com/prob-supp.html Problem 1. Recall Problem
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Math 172 B Class Project 1 Object: to download a specific mortality table from the Society of Actuaries website and utilize mortality rates to calculate life expectancies Steps: 1. Log onto www.soa.org 2. At the bottom of the opening page, select popular
School: UCLA
Course: Math
Math 31A Lecture 5 Summation Notation November 7, 2015 Summation notation is a short-hand way of writing sums of a lot of numbers when there is a pattern. For example, we could write 1 + 2 + 3 + + 100 as 100 j j=1 The symbol indicates that this is a sum.
School: UCLA
Course: Geometry A
Chapter4 TrianglesBasic Geometry InequalitiesinTriangles Anglesandtheiroppositesidesintrianglesarerelated.Infact,thisisoftenreflectedinthe labelingofanglesandsidesintriangleillustrations. Anglesandtheiroppositesidesareoften labeledwiththesameletter.Anuppe
School: UCLA
Course: Geometry A
Chapter4 TrianglesBasic Geometry CentersofTriangles Thefollowingareallpointswhichcanbeconsideredthecenterofatriangle. Centroid(Medians) Thecentroidistheintersectionofthethreemediansofatriangle.Amedianisa linesegmentdrawnfromavertextothemidpointofthelineop
School: UCLA
Course: Geometry A
Chapter3 ParallelandPerpendicularLines Geometry ProvingLinesareParallel Thepropertiesofparallellinescutbyatransversalcanbeusedtoprovetwolinesareparallel. CorrespondingAngles Iftwolinescutbyatransversalhavecongruentcorrespondingangles, thenthelinesareparal
School: UCLA
Course: Geometry A
Chapter4 TrianglesBasic Geometry TypesofTriangles Scalene AScaleneTrianglehas3sidesofdifferent lengths.Becausethesidesareof differentlengths,theanglesmustalsobe ofdifferentmeasures. Equilateral AnEquilateralTrianglehasall3sidesthe samelength(i.e.,c
School: UCLA
Course: Geometry A
Chapter3 ParallelandPerpendicularLines Geometry ParallelLinesandTransversals Transversal B A C Consecutive:referstoanglesthatare onthesamesideofthetransversal. D Alternate:referstoanglesthatareon oppositesidesofthetransversal. ParallelLines F E G Interior
School: UCLA
Course: Geometry A
Chapter2 Proofs Geometry Inductivevs.DeductiveReasoning InductiveReasoning Inductivereasoningusesobservationtoformahypothesisorconjecture.Thehypothesiscan thenbetestedtoseeifitistrue.Thetestmustbeperformedinordertoconfirmthe hypothesis. Example:Observetha
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Course: Geometry A
Chapter2 Proofs Geometry ConditionalStatements Aconditionalstatementcontainsbothahypothesisandaconclusioninthefollowingform: Ifhypothesis,thenconclusion. Foranyconditionalstatement,itispossibletocreatethreerelated conditionalstatements,asshownbelow.Inthet
School: UCLA
Course: Geometry A
Chapter1 BasicGeometry Geometry Angles PartsofanAngle Anangleconsistsoftworayswithacommon endpoint(or,initialpoint). Eachrayisasideoftheangle. Thecommonendpointiscalledthevertexof theangle. NamingAngles Anglescanbenamedinoneoftwoways: Pointvertexpointmeth
School: UCLA
Course: Geometry A
Chapter1 BasicGeometry Geometry DistanceBetweenPoints Distancemeasureshowfaraparttwothingsare.Thedistancebetweentwopointscanbe measuredinanynumberofdimensions,andisdefinedasthelengthofthelineconnectingthe twopoints.Distanceisalwaysapositivenumber. 1Dimens
School: UCLA
Course: Geometry A
Chapter1 BasicGeometry Geometry Points,Lines&Planes Item Illustration Notation Point Segment Ray Line lor Plane mor Definition Alocationinspace. Astraightpaththathastwoendpoints. Astraightpaththathasoneendpoint andextendsinfinitelyinonedirection. Astrai
School: UCLA
Course: Math
When using the normal distribution, choose the nearest z-value to find the probability, or if the probability is given, choose the nearest z-value. No interpolation should be used. Example: If the given z-value is 0.759, and you need to find Pr( Z < 0.759
School: UCLA
Course: CALCULUS OF A SINGLE VARIABLE
CALCULUS - CLUTCH CH.4: LIMITS (PART 1) ! ! www.clutchprep.com CALCULUS - CLUTCH CH.4: LIMITS (PART 1) INTRODUCTION TO LIMITS A limit is the _ . There are two types of limits, they are _ sided and _ sided. TWO- SIDED LIMITS Graphically A) B) C) ONE- SID
School: UCLA
Course: Linear Algebra
LINEAR ALGEBRA 4th STEPHEN H.FRIEDBERG, ARNOLD J.INSEL, LAWRENCE E.SPENCE Exercises Of Chapter 1-4 http : /math.pusan.ac.kr/caf e home/chuh/ Contents 1. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
School: UCLA
Course: CALCULUS FOR LIFE SCIENCES STUDENTS
3B Notes Sudesh Kalyanswamy 1 1.1 Chapter 6 Guiding Questions for the Chapter Here are some questions for you to ask yourself as you read. If you can answer them all by the end, youre in pretty good shape as far as understanding the material. After that,
School: UCLA
Course: Math 131a
(a) (5 points) State the Comparison :L Denition 2. The function f is said in he continuum; in 3 i Hunk", (b) (5 points) is the series E convergent? Justify you VT" 5 3 V5 > u 35 > 0 V1, 5 S Ii 7 ml < ,5 :5 Ix) , ru) < EL Fu- 1 ire-qu'nru' Inn}. 'll. wail:
School: UCLA
Course: Math32A
Remarks and a trick for the osculating plane Remark. Note that there is an error on Professor Taylors solution to Problem 5. The error occurs when he computes aN (1). The correct computation of aN (1) is aN (1) = (2, 1, 3) (2, 2, 1) = (0, 1, 2) NOT aN (1)
School: UCLA
Course: Math32A
About the 32A Final Location and Time Lecture #1 (the 10AM lecture) will be held in Moore 100 on Sunday, June 7th from 3:00PM until 6:00PM Lecture #2 (the 12PM lecture) will be held in Lakretz 110 on Sunday, June 7th from 3:00PM until 6:00PM Sections Co
School: UCLA
Course: Math32A
Math 21a: Multivariable calculus Distances overview DISTANCE POINT-POINT (3D). If P and Q are two points, then d(P, Q) = |P Q| is the distance between P and Q. We use the notation |v| instaed of |v| in this handout. DISTANCE POINT-PLANE (3D). If P is a po
School: UCLA
Course: CALCULUS FOR LIFE SCIENCE STUDENTS
i Mod-h BRZZ fevgewg Fined exam : 3/ 511/10 3 wade? V g«:>m a? Fm? 8 Lag _.o/20\3;. «,McuwiL-ztuct mgnmicmmxmmnlm WWW : E i ; E g i i; 'Yilxnc Mum wk 56:. at "moat 21,5 x . buggy THom I W: MiAVerm Km? *6 K) thiem. ' . mm mm he. 1-71.-?Ec§k\em.(3'3. at:
School: UCLA
Course: Introduction Of Complex Analysis
Math 132: Complex Analysis for Applications INSTRUCTOR: Kefeng Liu TIME and LOCATION: MWF 9:00-9:50AM, MS5117. OFFICE HOURS: TT 9:00-10:00AM in MS 6330 (Or by appointment) SYLLABUS: Official Syllabus Text: Complex Analysis by Tom Gamelin TA: Wenjian Liu,
School: UCLA
Course: Linear Algebra
Math 33A, Sec. 2 Linear Algebra and Applications Spring 2012 Instructor: Professor Alan J. Laub Dept. of Mathematics / Electrical Engineering Math Sciences 7945 phone: 310-825-4245 e-mail: laub@ucla.edu Lecture: MWF 2:00 p.m. 2:50 p.m.; Rolfe 1200 Office
School: UCLA
Course: Math
Dierential and Integral Calculus Math 31A, Lecture 5 Fall Quarter 2015 MWF 9:00am-9:50am, MS 6221 Instructor: Melissa Lynn mklynn@math.ucla.edu MS 3915B My Oce Hours: Mondays 11:30am-12:30pm, 3:30pm-4:30pm Fridays 10am-11am or by appointment Teaching Assi
School: UCLA
Course: CALCULUS OF A SINGLE VARIABLE
MATH 31A, Dierential and Integral Calculus, Lecture 4, Fall 2015 Exterior Course Website: http:/www.math.ucla.edu/heilman/31af15.html Prerequisite: Successful completion of Mathematics Diagnostic Test or course 1 with a grade of C- or better. Course Conte
School: UCLA
Course: Algebra-applicatns
v P y V ` V ` x G ~ y | G V | ` ` F G G y V F F GI | V U )HoASY2dddI)dY)2vdG u !A)w u zAdA)YV y V ` V ` x G ~ | G V | ` ` G V F GI X y V G ` ` FI ` ` 6 8 4 8 1 HAa92ddd!I dY)2vdG u F AHWdYaHa2 9!4 20 P U b U y b q b Uv b pv b gv b i tAACttAAmYACt7t
School: UCLA
Course: Math32A
UCLA Math 32A, Lecture 1 (Spring Quarter 2015) Calculus of Several Variables Instructor: David Wihr Taylor Time/Location: MWF 10:00am-10:50am in Rolfe 1200. Text: Calculus (2nd edition) by Jon Rogawski ISBN: 9781429294904 (Contact the UCLA Store in Ackerm
School: UCLA
Course: Math 31A
Dierential and Integral Calculus, Math 31a, Fall 2013, Lec 1 - Course Info Instructor: Tonci Antunovic, 6156 Math Sciences Building, tantunovic@math.ucla.edu, www.math.ucla.edu/tantunovic Instructor Oce Hours: Monday 11-12:30 and Thursday 5-6:30 in 6156 M
School: UCLA
Course: Sftwr-scntfc Cmpttn
Environment12: SustainabilityandtheEnvironment Summer2015 UCLAInstituteoftheEnvironment AndSustainability Page 1 of 5 SustainabilityandtheEnvironment Summer2015 Synopsis: There are a number of social, demographic, and ecological changes concurrently affec
School: UCLA
Course: SYSTMS-DIFFNTL EQTN
Math 131 A/1: Analysis, UCLA, Spring 2015 Instructor name Dr. Roman Taranets, Department of Mathematics, UCLA E-mail: taranets @ math.ucla.edu Website: www.math.ucla.edu/~taranets Office Phone: (310) 825-4746 Office: MS 7360 Office Hours: MWF 9:00 am
School: UCLA
Course: SYSTMS-DIFFNTL EQTN
MATH 135-2 Instructor: Sungjin Kim, MS 6617A, E-mail: i707107@math.ucla.edu Meeting Time and Location: MWF 3:00 - 3:50 PM, Room: MS 5200 O ce Hours: MW 11:45 PM - 12:30 PM, Room: MS 6617A Textbook: G. Simmons, Dierential Equations (2nd) with Applications
School: UCLA
Course: Differential Equations
Math 33B, Differential Equations Lecture 1, Spring 2015 Instructor: Will Conley Email: wconley@ucla.edu Office: MS 6322 Office Hours: Tentatively Wednesdays 3:004:00 Thursdays 12:001:30 and 3:004:00, and by appointment Website: http:/www.math.ucla.edu/~wc
School: UCLA
Course: TEXTBOOK SOLUTION
Math 174E, Lecture 1 (Winter 2014) Mathematics of Finance Instructor: David W. Taylor Time/Location: MWF 3:00pm-3:50pm in MS 5127. Text: Options, Futures, and Other Derivatives (8th edition) by John C. Hull. Instructor: David W. Taylor Oce Hours: Monday a
School: UCLA
Course: CALCULUS FOR LIFE SCIENCES STUDENTS
Math 3B Integral calculus and differential equations Prof. M. Roper Young Hall CS 76, MWF 2-2.50pm. All inquiries about this class should be posted to the class Piazza page. The only emails that I will respond to are questions about how to login to Piazza
School: UCLA
Course: Calculus
Sample Syllabus Philosophical Analysis of Contemporary Moral Issues Winter 2015, MW 8-9:50 Instructor: Robert Hughes (hughes@humnet.ucla.edu) Instructor office hours: Dodd 378, M 11-12 and W 10-11 or by appointment TAs: Kelsey Merrill (kmerrill@humnet.ucl
School: UCLA
Course: Math 170
Probability Theory, Math 170A, Winter 2015 - Course Info Instructor: Tonci Antunovic, 6156 Math Sciences Building, tantunovic@math.ucla.edu, www.math.ucla.edu/~tantunovic Instructor Oce Hours: Monday 10-12am and Wednesday 9-10am in 6156 Math Sciences Buil
School: UCLA
Course: Math 1
Math1 Fall2012 Lecture1(8am),Lecture2(10am)andLecture3(1pm) Instructor Paige Greene E-Mail: paige@math.ucla.edu Office Number: MS 6617 B Office Hours: Mon and Wed (copy from me in class) Course Goals: The purpose of Math 1 is to give you a strong preparat
School: UCLA
Course: Math 33B
Math33BDifferentialEquations Winter2015,lecture3 Time:MWF3:003:50p.m. Room:MS4000A Discussionsections: 3A T3:00P3:50PGEOLOGY6704 3B R3:00P3:50PGEOLOGY4645 Instructor:YuZhang Office:MS6617F Officehours:Monday2:003:00p.m. Wednesday4:006:00p.m. Email:yuzhang
School: UCLA
Course: CALCULUS FOR LIFE SCIENCES STUDENTS
Math 3B Integral calculus and differential equations Prof. M. Roper Young Hall CS 76, MWF 2-2.50pm. All inquiries about this class should be posted to the class Piazza page. The only emails that I will respond to are questions about how to login to Piazza
School: UCLA
Course: Actuarial Math
Math 172A: Syllabus, Structure, Details and Policies (Fall 2014) Introduction to Financial Mathematics This course is designed to provide an understanding of basic concepts of actuarial/financial mathematics. The course touches most (but not all) syllabus
School: UCLA
Course: Abstract Algebra
Math 32B (Calculus of several variables), Section 2 Fall 2013 MWF 2:00-2:50 pm, 1178 Franz Hall UCLA Instructor: William Simmons, MS 6617A, simmons@math.ucla.edu. Oce Hours (held in MS 6617A): M: 8:00-8:55 am T: 10:30-11:30 am W: 3:30-4:30 pm Teaching Ass
School: UCLA
Course: Linear Algebra
Math 115A, Linear Algebra, Lecture 6, Fall 2014 Exterior Course Website: http:/www.math.ucla.edu/heilman/115af14.html Prerequisite: MATH 33A, Linear algebra and applications. Course Content: Linear independence, bases, orthogonality, the Gram-Schmidt proc
School: UCLA
Course: Math32A
Math 32A, Calculus of Several Variables Lecture 1, Winter 2014 Instructor: Will Conley Email: wconley@ucla.edu Office: MS 6322 Office Hours: Tentatively Wednesdays 11:3012:30, Thursdays 1:302:30, and any other time by appointment. Website: http:/www.math.
School: UCLA
Course: Linear Algebra And Applications
Math 33A, Section 2 Linear Algebra and Applications Winter 2014 Instructor: Professor Alan J. Laub Dept. of Mathematics / Electrical Engineering Math Sciences 7945 phone: 310-825-4245 e-mail: laub@ucla.edu Lecture: MWF 1:00 p.m. 1:50 p.m. Math Sciences 40
School: UCLA
Course: Precalculus
Math1 Fall2012 Lecture1(8am),Lecture2(10am)andLecture3(1pm) Instructor Paige Greene E-Mail: paige@math.ucla.edu Office Number: MS 6617 B Office Hours: Mon and Wed (copy from me in class) Course Goals: The purpose of Math 1 is to give you a strong preparat
School: UCLA
Course: Math 33B
Differential Equations MATH 33B-1 Instructor Details: Lecture 1: Instructor: Office: Office hours: MWF 12PM-12:50PM in MOORE 100 Prof. Zachary Maddock MS 6236 (MS := Mathematical Sciences) Mondays 1:30PM-2:30PM Wednesdays 9:50AM-10:50AM, 1:30PM-2:30PM Tea
School: UCLA
Course: Linear Algebra
Course information and Policy, v. of Oct.1st, 2013 Marc-Hubert NICOLE Linear Algebra with Applications Math 33A-1, Lecture 1, Fall 2013. U.C.L.A. -Instructor Marc-Hubert NICOLE Lectures: MWF 8:00-8:50 AM in Rolfe 1200. Ofce: MS 5230 (see Ofce Hours belo
School: UCLA
Course: Linear Algebra
Winter 2011, UCLA Department of Economics Economics 106G: Introduction to Game Theory Instructor: Ichiro Obara Bunche 9381 E-mail: iobara@econ.ucla.edu Web: http:/www.econ.ucla.edu/iobara Office Hour: Tuesday 1:00pm - 2:30pm or by appointment. Time and Lo
School: UCLA
Course: Linear Algebra
Math 115A Fall 2012 Professor: David Gieseker Oce: Math Sciences 5636. Phone: 206-6321, email: dag at math ucla edu Oce Hours: M, W, F 9:30-10, 11-11:30. M,W 2-2:30 and by appointment. Text: Linear Algebra, Friedberg, Insel, Spence. Material to be covered
School: UCLA
Course: Math32A
S YLLABUS : P HYSICS 1B, L ECTURE 1, W INTER 2012 I NSTRUCTOR : Prof. Troy Carter 4-909 PAB (310) 825-4770 tcarter@physics.ucla.edu O FFICE H OURS : M 10-11AM, T 1PM-2PM, F 1-2PM, other times by appointment C OURSE A DMINISTRATOR : Elaine Dolalas (handles