MAT 4872 Midterm Review Spring 2007 Prof. S. Singh
|
|
- Elisabeth Bradley
- 5 years ago
- Views:
Transcription
1 MAT 487 Midterm Review Spring 007 Prof S Singh Answer all questions carefully and give references for material used from journals Sketch all state spaces for all transformations You are required to work indepently 1 Let X and Y be continuous random variables with densities f X ( x ) and f ( y ), respectively, and suppose that X and Y are Y indepent Use the transformation U = X + Y, V = X, to show that the density of U satisfies the convolution equation: fu ( u) = fx ( x) fy ( u v) dv If X and Y are iid uniform distributions on, find the distribution of U = X + Y (0,1) ( ) mod1 We will examine the quadratic equation Ax + Bx+ C, where A, B and C are indepent random variables, uniformly distributed on (0,1) a Find the distribution of X = lnb b Find the distribution of Y = ln A ln C c Explain why X and Y are indepent d Show that the probability that the quadratic equation have real roots is equivalent to solving PY ( X ln 4) e Evaluate: PY ( X ln 4) f Write a program to simulate the values of A, B and C Your program must count the number of times that your simulated triple, ( A, BC, ) generate real roots to the quadratic equation Run your program for ten thousand trials and compare the probability obtained with your answer in part e
2 X Let X N 3, Define Y1 = 5X1 Xand X Y1 Y = X + 3X3 Find the moment generating function of Y 4 The random variables and Y are iid, both with density f( y) = y y ( for 1, ) Y1 and zero elsewhere Find the joint density of and U, where U1 density of U 1 U 1 Y1 = Y + Y 1 and U = Y1 + Y Find the marginal 5 A bivariate distribution is always normal if its marginals are normal Prove or give a counterexample to this claim Justify all steps in this solution
3 Problems and solutions 1 Let X and Y be continuous random variables with densities f X ( x ) and f ( y ), respectively, and suppose that X and Y are Y indepent Use the transformation U = X + Y, V = X, to show that the density of U satisfies the convolution equation: fu ( u) = fx ( x ) fy ( u v) dv If X and Y are iid uniform distributions on ( 0,1 ), find the distribution of U = ( X + Y) mod1 The first part is a direct computation U is a uniform distribution on ( 0,1 ) Explain this carefully We will examine the quadratic equation Ax + Bx+ C, where A, B and C are indepent random variables, uniformly distributed on (0,1) g Find the distribution of X = lnb 1 x / X e for x ( 0, ) and zero elsewhere This is an exponential distribution h Find the distribution of Y = ln A ln C y Y ye for and zero elsewhere This is a gamma distribution y (0, ) i Explain why X and Y are indepent This is clear Explain j Show that the probability that the quadratic equation PY X ln 4 have real roots is equivalent to solving ( ) Start with B 4AC and take logs of both sides
4 k Evaluate: PY ( X ln 4) { xy, 0 x,0 y } The region ( ) region {( uv, ) 0 v, v u} U = Y X and V = Y u / e for u,0 9 f( u) = 1 u u+ e for u 0, 3 3 < < gets mapped into the < < by the transformation: ( ) ( ) ( X ln 4) = ( ln 4) PY 1 u 5 1 PU = u e du ln ln 4 + = l Write a program to simulate the values of A, B and C Your program must count the number of times that your simulated triple, ( A, BC, ) generate real roots to the quadratic equation Run your program for ten thousand trials and compare the probability obtained with your answer in part e The matlab program below simulates the triple ( A, BC, ) ; finds the probability of real roots; and gives a graphical illustration of the distribution of the real roots in red and the imaginary roots in green function qroot(n) counter=0; for i=1:n A=rand; B=rand; C=rand; D=B^-4*A*C; if (D>=0) counter=counter+1; plot(i,d,'r') hold on else plot(i,d,'g') hold on
5 probability = counter/n Probability= 0579 for 10,000 triples, which is close to the theoretical value Here is an alternative way to solve the above problem Solution II For imaginary roots: c > b /4a(See the shaded region above), if b is fixed, then b /4a < c< 1, b /4< a < 1 from the above diagram Clearly 0< b < 1, therefore the probability of real roots is:
6 b /4 b /4a 5 1 dc da db = + ln 36 6 The above solution is much more nicer and it can be exted to consider all quadratics If we consider ab, and c to be uniform on ( 1,1), then imaginary roots can be found when a and c have the same sign In the above analysis we consider the case a > 0 and c > 0 The joint density function of ab, and c is 1/ 8 on the cube of side units centered around the origin Now the probability of real roots is: dc da db = + ln b /4 b /4a A simulation run by matlab with the program qroot1 gives: qroot1( ) probability = function qroot1(n) counter=0; for i=1:n A=*rand-1; B=*rand-1; C=*rand-1; D=B^-4*A*C; if (D>=0) counter=counter+1; probability = counter/n X Let X N 3, Define Y1 = 5X1 Xand X Y1 Y = X + 3X3 Find the moment generating function of Y / 1 / M () t = exp t μ + t t) = exp(45t1 + 84t1t + 49t + 4t t 1 y1 y
7 4 The random variables and Y are iid, both with density f( y) = y ( for 1, y ) Y1 and zero elsewhere Find the joint density of and U, where U1 density of U 1 U 1 Y1 = Y + Y 1 and U = Y1 + Y Find the marginal f u ( u) 1 1 for u 0, 1 ( ) u = 1 1 for u,1 u 5 A bivariate distribution is always normal if its marginals are normal Prove or give a counterexample to this claim Justify all steps in this solution ( x + y ) xye 05( x + y ) Consider the joint density f( x, y) = e, π where x and y, for a counterexample
8
Stat 366 A1 (Fall 2006) Midterm Solutions (October 23) page 1
Stat 366 A1 Fall 6) Midterm Solutions October 3) page 1 1. The opening prices per share Y 1 and Y measured in dollars) of two similar stocks are independent random variables, each with a density function
More informationUNIT Define joint distribution and joint probability density function for the two random variables X and Y.
UNIT 4 1. Define joint distribution and joint probability density function for the two random variables X and Y. Let and represent the probability distribution functions of two random variables X and Y
More informationSTAT515, Review Worksheet for Midterm 2 Spring 2019
STAT55, Review Worksheet for Midterm 2 Spring 29. During a week, the proportion of time X that a machine is down for maintenance or repair has the following probability density function: 2( x, x, f(x The
More information2 Functions of random variables
2 Functions of random variables A basic statistical model for sample data is a collection of random variables X 1,..., X n. The data are summarised in terms of certain sample statistics, calculated as
More informationChapter 4 Multiple Random Variables
Review for the previous lecture Definition: n-dimensional random vector, joint pmf (pdf), marginal pmf (pdf) Theorem: How to calculate marginal pmf (pdf) given joint pmf (pdf) Example: How to calculate
More informationTwo hours. Statistical Tables to be provided THE UNIVERSITY OF MANCHESTER. 14 January :45 11:45
Two hours Statistical Tables to be provided THE UNIVERSITY OF MANCHESTER PROBABILITY 2 14 January 2015 09:45 11:45 Answer ALL four questions in Section A (40 marks in total) and TWO of the THREE questions
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationEssential Mathematics
Appendix B 1211 Appendix B Essential Mathematics Exponential Arithmetic Exponential notation is used to express very large and very small numbers as a product of two numbers. The first number of the product,
More informationSTAT 515 MIDTERM 2 EXAM November 14, 2018
STAT 55 MIDTERM 2 EXAM November 4, 28 NAME: Section Number: Instructor: In problems that require reasoning, algebraic calculation, or the use of your graphing calculator, it is not sufficient just to write
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 05 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA6 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM004 REGULATION : R008 UPDATED ON : Nov-Dec 04 (Scan
More informationContinuous Random Variables
1 / 24 Continuous Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 27, 2013 2 / 24 Continuous Random Variables
More informationSTA2603/205/1/2014 /2014. ry II. Tutorial letter 205/1/
STA263/25//24 Tutorial letter 25// /24 Distribution Theor ry II STA263 Semester Department of Statistics CONTENTS: Examination preparation tutorial letterr Solutions to Assignment 6 2 Dear Student, This
More informationMath 113 Winter 2013 Prof. Church Midterm Solutions
Math 113 Winter 2013 Prof. Church Midterm Solutions Name: Student ID: Signature: Question 1 (20 points). Let V be a finite-dimensional vector space, and let T L(V, W ). Assume that v 1,..., v n is a basis
More informationUCSD ECE 153 Handout #20 Prof. Young-Han Kim Thursday, April 24, Solutions to Homework Set #3 (Prepared by TA Fatemeh Arbabjolfaei)
UCSD ECE 53 Handout #0 Prof. Young-Han Kim Thursday, April 4, 04 Solutions to Homework Set #3 (Prepared by TA Fatemeh Arbabjolfaei). Time until the n-th arrival. Let the random variable N(t) be the number
More informationIEOR 4703: Homework 2 Solutions
IEOR 4703: Homework 2 Solutions Exercises for which no programming is required Let U be uniformly distributed on the interval (0, 1); P (U x) = x, x (0, 1). We assume that your computer can sequentially
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions
More information[Chapter 6. Functions of Random Variables]
[Chapter 6. Functions of Random Variables] 6.1 Introduction 6.2 Finding the probability distribution of a function of random variables 6.3 The method of distribution functions 6.5 The method of Moment-generating
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 08 SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : University Questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) (Scan the
More informationIntroduction to Differentials
Introduction to Differentials David G Radcliffe 13 March 2007 1 Increments Let y be a function of x, say y = f(x). The symbol x denotes a change or increment in the value of x. Note that a change in the
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. 60 minutes.
Closed book and notes. 60 minutes. A summary table of some univariate continuous distributions is provided. Four Pages. In this version of the Key, I try to be more complete than necessary to receive full
More informationChapter 3 Exponential and Logarithmic Functions
Chapter 3 Exponential and Logarithmic Functions Overview: 3.1 Exponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Exponential and
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationChapter 4 Multiple Random Variables
Review for the previous lecture Theorems and Examples: How to obtain the pmf (pdf) of U = g ( X Y 1 ) and V = g ( X Y) Chapter 4 Multiple Random Variables Chapter 43 Bivariate Transformations Continuous
More informationThe final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.
Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic
More informationCourse: ESO-209 Home Work: 1 Instructor: Debasis Kundu
Home Work: 1 1. Describe the sample space when a coin is tossed (a) once, (b) three times, (c) n times, (d) an infinite number of times. 2. A coin is tossed until for the first time the same result appear
More informationThis exam contains 6 questions. The questions are of equal weight. Print your name at the top of this page in the upper right hand corner.
GROUND RULES: This exam contains 6 questions. The questions are of equal weight. Print your name at the top of this page in the upper right hand corner. This exam is closed book and closed notes. Show
More informationInstructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.
Exam 3 Math 850-007 Fall 04 Odenthal Name: Instructions: No books. No notes. Non-graphing calculators only. You are encouraged, although not required, to show your work.. Evaluate the iterated integral
More informationFormulas for probability theory and linear models SF2941
Formulas for probability theory and linear models SF2941 These pages + Appendix 2 of Gut) are permitted as assistance at the exam. 11 maj 2008 Selected formulae of probability Bivariate probability Transforms
More informationLesson 9 Exploring Graphs of Quadratic Functions
Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point
More informationThis exam is closed book and closed notes. (You will have access to a copy of the Table of Common Distributions given in the back of the text.
TEST #3 STA 5326 December 4, 214 Name: Please read the following directions. DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Directions This exam is closed book and closed notes. (You will have access to
More informationJoint Distributions. (a) Scalar multiplication: k = c d. (b) Product of two matrices: c d. (c) The transpose of a matrix:
Joint Distributions Joint Distributions A bivariate normal distribution generalizes the concept of normal distribution to bivariate random variables It requires a matrix formulation of quadratic forms,
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More information4. CONTINUOUS RANDOM VARIABLES
IA Probability Lent Term 4 CONTINUOUS RANDOM VARIABLES 4 Introduction Up to now we have restricted consideration to sample spaces Ω which are finite, or countable; we will now relax that assumption We
More informationP 1.5 X 4.5 / X 2 and (iii) The smallest value of n for
DHANALAKSHMI COLLEGE OF ENEINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING MA645 PROBABILITY AND RANDOM PROCESS UNIT I : RANDOM VARIABLES PART B (6 MARKS). A random variable X
More informationProblem Y is an exponential random variable with parameter λ = 0.2. Given the event A = {Y < 2},
ECE32 Spring 25 HW Solutions April 6, 25 Solutions to HW Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics where
More informationMATH 360. Probablity Final Examination December 21, 2011 (2:00 pm - 5:00 pm)
Name: MATH 360. Probablity Final Examination December 21, 2011 (2:00 pm - 5:00 pm) Instructions: The total score is 200 points. There are ten problems. Point values per problem are shown besides the questions.
More informationMathematics 426 Robert Gross Homework 9 Answers
Mathematics 4 Robert Gross Homework 9 Answers. Suppose that X is a normal random variable with mean µ and standard deviation σ. Suppose that PX > 9 PX
More informationHW4 : Bivariate Distributions (1) Solutions
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 7 Néhémy Lim HW4 : Bivariate Distributions () Solutions Problem. The joint probability mass function of X and Y is given by the following table : X Y
More informationMTH135/STA104: Probability
MTH5/STA4: Probability Homework # Due: Tuesday, Dec 6, 5 Prof Robert Wolpert Three subjects in a medical trial are given drug A After one week, those that do not respond favorably are switched to drug
More informationMath 175 Common Exam 2A Spring 2018
Math 175 Common Exam 2A Spring 2018 Part I: Short Form The first seven (7) pages are short answer. You don t need to show work. Partial credit will be rare and small. 1. (8 points) Suppose f(x) is a function
More informationTHE QUEEN S UNIVERSITY OF BELFAST
THE QUEEN S UNIVERSITY OF BELFAST 0SOR20 Level 2 Examination Statistics and Operational Research 20 Probability and Distribution Theory Wednesday 4 August 2002 2.30 pm 5.30 pm Examiners { Professor R M
More informationRYERSON UNIVERSITY DEPARTMENT OF MATHEMATICS MTH 514 Stochastic Processes
RYERSON UNIVERSITY DEPARTMENT OF MATHEMATICS MTH 514 Stochastic Processes Midterm 2 Assignment Last Name (Print):. First Name:. Student Number: Signature:. Date: March, 2010 Due: March 18, in class. Instructions:
More information9.07 Introduction to Probability and Statistics for Brain and Cognitive Sciences Emery N. Brown
9.07 Introduction to Probability and Statistics for Brain and Cognitive Sciences Emery N. Brown I. Objectives Lecture 5: Conditional Distributions and Functions of Jointly Distributed Random Variables
More informationChapter 3 Exponential and Logarithmic Functions
Chapter 3 Exponential and Logarithmic Functions Overview: 3.1 Exponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Exponential and
More information2 Random Variable Generation
2 Random Variable Generation Most Monte Carlo computations require, as a starting point, a sequence of i.i.d. random variables with given marginal distribution. We describe here some of the basic methods
More informationLogarithmic and Exponential Equations and Change-of-Base
Logarithmic and Exponential Equations and Change-of-Base MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to solve exponential equations
More informationComputer Vision & Digital Image Processing. Periodicity of the Fourier transform
Computer Vision & Digital Image Processing Fourier Transform Properties, the Laplacian, Convolution and Correlation Dr. D. J. Jackson Lecture 9- Periodicity of the Fourier transform The discrete Fourier
More informationPaper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours
1. Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Mark scheme Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question
More informationClosed book and notes. 60 minutes. Cover page and four pages of exam. No calculators.
IE 230 Seat # Closed book and notes. 60 minutes. Cover page and four pages of exam. No calculators. Score Exam #3a, Spring 2002 Schmeiser Closed book and notes. 60 minutes. 1. True or false. (for each,
More informationConditional Distributions
Conditional Distributions The goal is to provide a general definition of the conditional distribution of Y given X, when (X, Y ) are jointly distributed. Let F be a distribution function on R. Let G(,
More informationAre you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*
Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course
More informationUC Berkeley Department of Electrical Engineering and Computer Sciences. EECS 126: Probability and Random Processes
UC Berkeley Department of Electrical Engineering and Computer Sciences EECS 6: Probability and Random Processes Problem Set 3 Spring 9 Self-Graded Scores Due: February 8, 9 Submit your self-graded scores
More informationElementary ODE Review
Elementary ODE Review First Order ODEs First Order Equations Ordinary differential equations of the fm y F(x, y) () are called first der dinary differential equations. There are a variety of techniques
More informationMATH c UNIVERSITY OF LEEDS Examination for the Module MATH2715 (January 2015) STATISTICAL METHODS. Time allowed: 2 hours
MATH2750 This question paper consists of 8 printed pages, each of which is identified by the reference MATH275. All calculators must carry an approval sticker issued by the School of Mathematics. c UNIVERSITY
More informationIf a function has an inverse then we can determine the input if we know the output. For example if the function
1 Inverse Functions We know what it means for a relation to be a function. Every input maps to only one output, it passes the vertical line test. But not every function has an inverse. A function has no
More informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1
More informationRandom Variables and Probability Distributions
CHAPTER Random Variables and Probability Distributions Random Variables Suppose that to each point of a sample space we assign a number. We then have a function defined on the sample space. This function
More informationOrder Statistics and Distributions
Order Statistics and Distributions 1 Some Preliminary Comments and Ideas In this section we consider a random sample X 1, X 2,..., X n common continuous distribution function F and probability density
More informationPRE-LEAVING CERTIFICATE EXAMINATION, 2010
L.7 PRE-LEAVING CERTIFICATE EXAMINATION, 00 MATHEMATICS HIGHER LEVEL PAPER (300 marks) TIME : ½ HOURS Attempt SIX QUESTIONS (50 marks each). WARNING: Marks will be lost if all necessary work is not clearly
More information4th Quarter Pacing Guide LESSON PLANNING
Domain: Reasoning with Equations and Inequalities Cluster: Solve equations and inequalities in one variable KCAS: A.REI.4a: Solve quadratic equations in one variable. a. Use the method of completing the
More informationQualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama
Qualifying Exam CS 661: System Simulation Summer 2013 Prof. Marvin K. Nakayama Instructions This exam has 7 pages in total, numbered 1 to 7. Make sure your exam has all the pages. This exam will be 2 hours
More informationMultiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More informationChapter 5 continued. Chapter 5 sections
Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationMidterm Preparation Problems
Midterm Preparation Problems The following are practice problems for the Math 1200 Midterm Exam. Some of these may appear on the exam version for your section. To use them well, solve the problems, then
More informationSolutions to Homework 11
Solutions to Homework 11 Read the statement of Proposition 5.4 of Chapter 3, Section 5. Write a summary of the proof. Comment on the following details: Does the proof work if g is piecewise C 1? Or did
More informationCore Mathematics C4 Advanced Level
Paper Reference(s) 6666/0 Edexcel GCE Core Mathematics C4 Advanced Level Thursday June 0 Afternoon Time: hour 0 minutes Materials required for examination Mathematical Formulae (Pink) Items included with
More informationNotes for Math 324, Part 19
48 Notes for Math 324, Part 9 Chapter 9 Multivariate distributions, covariance Often, we need to consider several random variables at the same time. We have a sample space S and r.v. s X, Y,..., which
More informationBivariate Transformations
Bivariate Transformations October 29, 29 Let X Y be jointly continuous rom variables with density function f X,Y let g be a one to one transformation. Write (U, V ) = g(x, Y ). The goal is to find the
More informationCore Mathematics C34
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C34 Advanced Tuesday 20 June 2017 Afternoon Time: 2 hours 30 minutes
More informationExam 2. Jeremy Morris. March 23, 2006
Exam Jeremy Morris March 3, 006 4. Consider a bivariate normal population with µ 0, µ, σ, σ and ρ.5. a Write out the bivariate normal density. The multivariate normal density is defined by the following
More informationECE 313: Conflict Final Exam Tuesday, May 13, 2014, 7:00 p.m. 10:00 p.m. Room 241 Everitt Lab
University of Illinois Spring 1 ECE 313: Conflict Final Exam Tuesday, May 13, 1, 7: p.m. 1: p.m. Room 1 Everitt Lab 1. [18 points] Consider an experiment in which a fair coin is repeatedly tossed every
More informationHomework 10 (due December 2, 2009)
Homework (due December, 9) Problem. Let X and Y be independent binomial random variables with parameters (n, p) and (n, p) respectively. Prove that X + Y is a binomial random variable with parameters (n
More informationThis midterm covers Chapters 6 and 7 in WMS (and the notes). The following problems are stratified by chapter.
This midterm covers Chapters 6 and 7 in WMS (and the notes). The following problems are stratified by chapter. Chapter 6 Problems 1. Suppose that Y U(0, 2) so that the probability density function (pdf)
More informationMidterm 1. Every element of the set of functions is continuous
Econ 200 Mathematics for Economists Midterm Question.- Consider the set of functions F C(0, ) dened by { } F = f C(0, ) f(x) = ax b, a A R and b B R That is, F is a subset of the set of continuous functions
More informationPerhaps the simplest way of modeling two (discrete) random variables is by means of a joint PMF, defined as follows.
Chapter 5 Two Random Variables In a practical engineering problem, there is almost always causal relationship between different events. Some relationships are determined by physical laws, e.g., voltage
More informationASM Study Manual for Exam P, Second Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA Errata
ASM Study Manual for Exam P, Second Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA (krzysio@krzysio.net) Errata Effective July 5, 3, only the latest edition of this manual will have its errata
More informationMath 151. Rumbos Spring Solutions to Review Problems for Exam 2
Math 5. Rumbos Spring 22 Solutions to Review Problems for Exam 2. Let X and Y be independent Normal(, ) random variables. Put Z = Y X. Compute the distribution functions (z) and (z). Solution: Since X,
More informationDiff. Eq. App.( ) Midterm 1 Solutions
Diff. Eq. App.(110.302) Midterm 1 Solutions Johns Hopkins University February 28, 2011 Problem 1.[3 15 = 45 points] Solve the following differential equations. (Hint: Identify the types of the equations
More informationRandom Signals and Systems. Chapter 3. Jitendra K Tugnait. Department of Electrical & Computer Engineering. Auburn University.
Random Signals and Systems Chapter 3 Jitendra K Tugnait Professor Department of Electrical & Computer Engineering Auburn University Two Random Variables Previously, we only dealt with one random variable
More informationAdd Math (4047/02) Year t years $P
Add Math (4047/0) Requirement : Answer all questions Total marks : 100 Duration : hour 30 minutes 1. The price, $P, of a company share on 1 st January has been increasing each year from 1995 to 015. The
More information24. Partial Differentiation
24. Partial Differentiation The derivative of a single variable function, d f(x), always assumes that the independent variable dx is increasing in the usual manner. Visually, the derivative s value at
More informationMA Spring 2013 Lecture Topics
LECTURE 1 Chapter 12.1 Coordinate Systems Chapter 12.2 Vectors MA 16200 Spring 2013 Lecture Topics Let a,b,c,d be constants. 1. Describe a right hand rectangular coordinate system. Plot point (a,b,c) inn
More informationAP Calculus BC Chapter 4 AP Exam Problems. Answers
AP Calculus BC Chapter 4 AP Exam Problems Answers. A 988 AB # 48%. D 998 AB #4 5%. E 998 BC # % 5. C 99 AB # % 6. B 998 AB #80 48% 7. C 99 AB #7 65% 8. C 998 AB # 69% 9. B 99 BC # 75% 0. C 998 BC # 80%.
More informationECE 650 1/11. Homework Sets 1-3
ECE 650 1/11 Note to self: replace # 12, # 15 Homework Sets 1-3 HW Set 1: Review Assignment from Basic Probability 1. Suppose that the duration in minutes of a long-distance phone call is exponentially
More informationProblem Set. Assignment #1. Math 3350, Spring Feb. 6, 2004 ANSWERS
Problem Set Assignment #1 Math 3350, Spring 2004 Feb. 6, 2004 ANSWERS i Problem 1. [Section 1.4, Problem 4] A rocket is shot straight up. During the initial stages of flight is has acceleration 7t m /s
More informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationThis exam contains 13 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability and Statistics FS 2017 Session Exam 22.08.2017 Time Limit: 180 Minutes Name: Student ID: This exam contains 13 pages (including this cover page) and 10 questions. A Formulae sheet is provided
More informationMath 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours
Math 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours Name: Please read the questions carefully. You will not be given partial credit on the basis of having misunderstood a question, and please
More information2001 Higher Maths Non-Calculator PAPER 1 ( Non-Calc. )
001 PAPER 1 ( Non-Calc. ) 1 1) Find the equation of the straight line which is parallel to the line with equation x + 3y = 5 and which passes through the point (, 1). Parallel lines have the same gradient.
More informationSolutions to Assignment #8 Math 501 1, Spring 2006 University of Utah
Solutions to Assignment #8 Math 5, Spring 26 University of Utah Problems:. A man and a woman agree to meet at a certain location at about 2:3 p.m. If the man arrives at a time uniformly distributed between
More informationEXAM 3 MAT 167 Calculus I Spring is a composite function of two functions y = e u and u = 4 x + x 2. By the. dy dx = dy du = e u x + 2x.
EXAM MAT 67 Calculus I Spring 20 Name: Section: I Each answer must include either supporting work or an explanation of your reasoning. These elements are considered to be the main part of each answer and
More informationMath 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
More informationStatistics Ph.D. Qualifying Exam: Part I October 18, 2003
Statistics Ph.D. Qualifying Exam: Part I October 18, 2003 Student Name: 1. Answer 8 out of 12 problems. Mark the problems you selected in the following table. 1 2 3 4 5 6 7 8 9 10 11 12 2. Write your answer
More informationMultivariate Distributions
IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Multivariate Distributions We will study multivariate distributions in these notes, focusing 1 in particular on multivariate
More informationDRAFT. Mathematical Methods 2017 Sample paper. Question Booklet 1
1 South Australian Certificate of Education Mathematical Methods 017 Sample paper Question Booklet 1 Part 1 Questions 1 to 10 Answer all questions in Part 1 Write your answers in this question booklet
More informationMultiple Random Variables
Multiple Random Variables Joint Probability Density Let X and Y be two random variables. Their joint distribution function is F ( XY x, y) P X x Y y. F XY ( ) 1, < x
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 03 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA 6 MATERIAL NAME : Problem Material MATERIAL CODE : JM08AM008 (Scan the above QR code for the direct download of
More informationMath Algebra I. PLD Standard Minimally Proficient Partially Proficient Proficient Highly Proficient. student
PLD Standard Minimally Proficient Partially Proficient Proficient Highly Proficient The Minimally Proficient student The Partially Proficient student The Proficient student The Highly Proficient student
More information0, otherwise. U = Y 1 Y 2 Hint: Use either the method of distribution functions or a bivariate transformation. (b) Find E(U).
1. Suppose Y U(0, 2) so that the probability density function (pdf) of Y is 1 2, 0 < y < 2 (a) Find the pdf of U = Y 4 + 1. Make sure to note the support. (c) Suppose Y 1, Y 2,..., Y n is an iid sample
More informationFinal exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 3B, Spring 207 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in the
More information