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School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 11 Solution Triet M. Le Page 107: 3(a): Let P be a polynomial of odd degree. One sees that limx- P (x) = - and limx P (x) = . Let a, b R such that P (a) < 0 and P (b) > 0. Clearly, P is continuous on [a, b]. By the Interm
School: Yale
Course: Optimization I
Answers to assignment due on October 14th. 2. Program 1D: z * = min y b, subject to yA c. To place this thing in the format of Program 12.1, we need to express it as a maximization problem with nonnegative variables and equality constraints. Lets replace
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 3: Measurable Functions and Their Unexpected Properties The due date for this assignment is 9/30 1. Consider the Cantor function : [0; 1] P [0; 1]. It is dened as follows: For each ! n x 2 C , the Cantor
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 1: -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0; 1] which admit decimal expansions such that the digits 2; 4; 6; 8 all appear at lea
School: Yale
QSO 510 QUANTITATIVE ANALYSIS LECTURE 2 Lingling Wang Outline Distributions in Business Applications Sampling Distribution Define sampling both with and without replacement. Identify a sampling distribution. Explain the relationship between the distributi
School: Yale
QSO 510 QUANTITATIVE ANALYSIS LECTURE 3 Lingling Wang QSO 510 2014 Last Lecture: Sampling Distribution Suppose, we know Population Parameters: and Population distribution What we want to know Sample statistics (sample mean) Parameters: and Sampling di
School: Yale
Course: Integral Calculus
Yale University, Department of Mathematics Math 115 Calculus Fall 2013 Final Exam Review Guide When/Where. The Final Exam will take place during 7:00 10:30 pm on Sunday, December 15th, 2013 in Davies Auditorium, located underground between Dunham Labs and
School: Yale
Week 7 Important Discrete Distributions Expectation and Variance Lecture 17. Bernoulli and Binomial Distributions. Expectation Revisited. Discrete Uniform Distribution. We have seen some examples that all outcomes of an experiment are equally likely. Let
School: Yale
Week 5 Lecture 11. Conditional distribution and Conditional density. Review. Discrete random variables X1 ; X2 ; ; Xn are mutually independent if P (X1 = x1 ; X2 = x2 ; ; Xn = xn ) = P (X1 = x1 ) P (X2 = x2 ) : : : P (Xn = xn ) . Continuous random variabl
School: Yale
Week 4 Lecture 8. Discrete conditional distribution. Examples 1: A doctor gives a patient a test for a particular cancer. Before the results of the test, the only evidence the doctor has to go on is that 1 woman in 1000 has this cancer. Experience has sho
School: Yale
Week 6 Lecture 14 Random Walks Drunkard Walk. Imagine now a drunkard walking randomly in an ideals ized 1 dimensional city ( or 2 dimensional, or 3 and higher dimensional city). The city is eectively innite and arranged in a 1 dimensional equally-spaced g
School: Yale
Week 3 Lecture 5. Expectation Probability Density Function: Let f (x) 0 and P (E ) as following Z P (X 2 E ) = f (x) dx. R f (x) dx = 1. Dene E Are the probability axioms satised? It is important to observe that there a similar paradox in the calculus Za
School: Yale
Week 2 Lecture 3. Expectation and Probability axioms. Random variable. A random variable is a real-valued function dened on the sample space, i.e., X (! ) is a function from to R. For example, for = fBB; BG; GB; GGg, your X could be the number of boys, th
School: Yale
Course: CalcOfFunctionsOfOneVariableII
SOLUTIONS TO EXAM 2, MATH 115, FALL 2012 1. (65 points) The graphs of f (x), g(x), and h(x) are shown below. You may assume that as x , the graphs of f , g, and h continue in a fashion similar to the trends observed in the graphs. 3 gx 2 1 hx f x 0 1 2 3
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 April 10th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor w
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 November 13th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructo
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Mathematics 115b Mid-term # 1 2/19/08 Name: Section: Please answer all SIX questions; each question is worth TEN points. You may use the table of integrals supplied with the test. However, books, notes, calculators, computers, cellphones may NOT be used.
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 14, 2009 1. Let T be the linear transformation from R3 to R3 which consists of rst rotating 30 degrees about the z axis and then rotating 90 degrees about the x axis. (I havent specied the directions of the rotations. Use whic
School: Yale
Course: Calculus
Math 1B - Fall 2006 10/09/2006 Integrals sin2 x cos2 x dx Use 2 sin x cos x = sin 2x, then use Ex 1. Ex 2. 0 1 (1 cos 4x) = sin2 2x. 2 4 dx 1 1 dx Try x = tan , dx = sec2 d. 2+1 4x 2 2 Ex 3. Ex 4. 4 dx 4 + e2x x Use Weierstrass substitutions, let t = tan
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 21, 2009 1. Find the inverse of 1 A = 3 2 2 4 , 4 0 1 3 using both row reduction and the adjugate. Check that you get the same answer. A1 8 = 10 3 4 3 2 7 2 1 1 1 2 2. Use the adjugate to nd a formula for the inverse of a 2 2
School: Yale
STAT 241/541, Probability Theory with Applications Fall 2013 Instructor: Harrison H. Zhou (huibin.zhou@yale.edu) O ce hours: Wednesday 4:00-6:00pm (tentative) or by appointments, Room 204, 24 Hillhouse Ave., James Dwight Dana House. T.A.: Corey Brier <cor