# Math 461 Fall 2006 Test 2 Solutions

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two formulas, by directly using the exponential density function. (a) P (0 < X < b) 1 e λb Solution. The exponential density is f(x) λe λx for x 0, and f(x) 0 for x < 0. We integrate the density to evaluate the probability: P (0 < X < b) b 0 b 0 f(x) dx λe λx dx e λx b 0 1 e λb. (b) E[X] 1/λ Solution. E[X] 0 xf(x) dx xλe λx dx 1 ye y dy λ 0 1 (y( e y ) ) 0 λ ( e y ) dy 0 1 ( ) 0 + e y dy λ 1 λ. 0 where y λx and dy λdx, by integration by parts

2 2. [15 points] An item is manufactured so that its width is normally distributed with mean µ 900 units and standard deviation units. What is the largest allowable value of such that at least 99% of the items will have widths in the range from 895 to 905 units? Solution. Write X Normal(900, 2 ) for the width. We want.99 P (895 < X < 905) ( P < X 900 ) < ( P 5 < Z < 5 ) where Z is standard normal Φ(5/) Φ( 5/) 2Φ(5/) 1 (why?). Rearranging the inequality, we want.995 Φ(5/), and after consulting our Table of the standard normal distribution, we see the inequality holds provided /, or 5/ Hence the largest allowable value of is 1.94 units. Remarks. 1. There is no continuity correction needed in this problem, because we are not approximating with a Normal random variable the random variable is already Normal! 2. This was a homework problem.

3 3. [15 points] Let X Uniform(0, 1). Find the density function of Y e X. Solution. First we determine which values Y can take on: X takes values between 0 and 1, and so Y e X takes values between e 0 1 and e 1 e. For any a and b in that range, 1 < a b < e, we have P (a < Y < b) P (a < e X < b) P (log a < X < log b) log b log a since X is uniform on (0, 1) F (b) F (a) where F (y) log y b a b a F (y) dy 1 y dy by the Fundamental Theorem since F (y) 1/y. Thus the density of Y is g(y) 1/y for 1 < y < e, and g(y) 0 otherwise. Remarks. 1. This problem was recommended as preparation on the Test 2 handout. 2. The general formula for transforming a density from X to Y is g(y) f(x) dx dy. In this problem y e x, and so x log y and dx 1 ; also we know dy y f(x) 1 if 0 < x e y < 1, and so we arrive at the same answer as above. But this general formula can be tricky to apply in practice, especially when more than one x value gives the same y value (e.g. y x 2 ). So it is better to argue directly, like we did in the Solution above.

4 4. [15 points] Suppose X Geometric(p), and P (X > k) 1/2. Show k log 2/ log(1/(1 p)). Solution. Recall X represents the number of trials for the first success, in an infinite sequence of Bernoulli trials. So X > k means precisely that the first k trials are failures. This occurs with probability (1 p) k, and hence 1 2 Rearranging the inequality gives P (X > k) (1 p) k. ( ) k p Taking logarithms gives ( ) 1 k log log 2. 1 p Notice 1/(1 p) is greater than 1, and so its logarithm is positive. Dividing out by that logarithm gives us an inequality on the k-values: k log 2 log(1/(1 p)). Remark. The formula P (X k) (1 p) k 1 was on the discrete densities handout. In this problem we have used it with k replaced by k + 1.

6 6. [25 points] An airplane has 85 seats for passengers. The airline has sold 100 tickets, but each passenger has only an 80% chance of showing up for the flight. (We assume passengers travel alone, and are independent of one another.) Find the approximate probability that every passenger who shows up will get a seat on the airplane. Solution. Write X for the number of passengers who show up, so that X Binomial(100,.80) because there are 100 passengers and we regard each one as a Bernoulli trial, with success meaning they show up for the flight. The mean and variance of X are µ np 100(.8) 80, 2 np(1 p) 100(.8)(.2) 16. The Binomial density is difficult to work with (by hand) because the numbers are so large, and so instead we use the Normal approximation (which is valid since n is large and 2 > 10). We want P (X 85) P (X 85.5) ( X µ P ( P Z 5.5 ) 16 P (Z 1.375) µ using the continuity correction ) where Z is standard normal Incidentally, using the original Binomial random variable gives an answer (by computer) of about.920. Remark. The Poisson approximation is not suitable for this problem, because p.8 is large here.

7 7. [Extra credit, 20 points] An airplane has s seats for passengers. The airline has sold n tickets, but each passenger has only probability p of showing up for the flight. (We assume passengers travel alone, and are independent of one another.) Write r for the price of each ticket, c for the cost (to the airline) of each flight, and d for the cost (to the airline) of each passenger who shows up but cannot get a seat on the airplane. Let X denote the number of passengers who show up (a random variable). Then profit nr c d(x s) + where t + t if t > 0 and t + 0 if t 0. (a) Evaluate the expected profit, using the Normal approximation. Solution. We have X Binomial(n, p) because there are n passengers and we regard each one as a Bernoulli trial, with success meaning they show up for the flight. The mean and variance of X are Therefore by linearity of expectation, µ np, 2 np(1 p). E[profit] nr c de[(x s) + ] [( X µ nr c de s µ ) ] + [( nr c de Z s µ ) ] by the Normal approximation to the Binomial ( nr c d z s µ ) φ(z)dz where φ(z) is the density function for the standard normal random variable. When evaluating the integral, notice we only really need to integrate from z s µ to z. Aside. We should really do a continuity correction in the above calculation. Where should it go? + + (b) Explain in principle how the airline can determine the best number n of seats to sell.

8 Solution. The airline should try to choose n to maximize the expected profit. They could do this by evaluating the above formula for each n, or else by regarding n as a real variable (rather than an integer variable) and using calculus methods to find the maximum: differentiate the expected profit with respect to n and find the critical points, then check which critical point is the maximum, and then round n to the nearest integer.

### Practice Problems #4

Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions

Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

### e.g. arrival of a customer to a service station or breakdown of a component in some system.

Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be

### Homework 4 - KEY. Jeff Brenion. June 16, 2004. Note: Many problems can be solved in more than one way; we present only a single solution here.

Homework 4 - KEY Jeff Brenion June 16, 2004 Note: Many problems can be solved in more than one way; we present only a single solution here. 1 Problem 2-1 Since there can be anywhere from 0 to 4 aces, the

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 5 Solutions

Math 370/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 5 Solutions About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the

### Aggregate Loss Models

Aggregate Loss Models Chapter 9 Stat 477 - Loss Models Chapter 9 (Stat 477) Aggregate Loss Models Brian Hartman - BYU 1 / 22 Objectives Objectives Individual risk model Collective risk model Computing

### Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008

Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the

### Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

### The Exponential Distribution

21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding

### Statistics 100A Homework 8 Solutions

Part : Chapter 7 Statistics A Homework 8 Solutions Ryan Rosario. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, the one-half

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### ECE302 Spring 2006 HW4 Solutions February 6, 2006 1

ECE302 Spring 2006 HW4 Solutions February 6, 2006 1 Solutions to HW4 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

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions

Math 70, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...

MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................

### STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE

STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE TROY BUTLER 1. Random variables and distributions We are often presented with descriptions of problems involving some level of uncertainty about

### Master s Theory Exam Spring 2006

Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem

### Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

### Math/Stats 342: Solutions to Homework

Math/Stats 342: Solutions to Homework Steven Miller (sjm1@williams.edu) November 17, 2011 Abstract Below are solutions / sketches of solutions to the homework problems from Math/Stats 342: Probability

### ), 35% use extra unleaded gas ( A

. At a certain gas station, 4% of the customers use regular unleaded gas ( A ), % use extra unleaded gas ( A ), and % use premium unleaded gas ( A ). Of those customers using regular gas, onl % fill their

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### ECE302 Spring 2006 HW3 Solutions February 2, 2006 1

ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 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

### 12.5: CHI-SQUARE GOODNESS OF FIT TESTS

125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability

### 1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

### Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions)

Math 370, Actuarial Problemsolving Spring 008 A.J. Hildebrand Practice Test, 1/8/008 (with solutions) About this test. This is a practice test made up of a random collection of 0 problems from past Course

### Final Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin

Final Mathematics 51, Section 1, Fall 24 Instructor: D.A. Levin Name YOU MUST SHOW YOUR WORK TO RECEIVE CREDIT. A CORRECT ANSWER WITHOUT SHOWING YOUR REASONING WILL NOT RECEIVE CREDIT. Problem Points Possible

### MAS108 Probability I

1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions

Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

### ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME

ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME Alexey Chuprunov Kazan State University, Russia István Fazekas University of Debrecen, Hungary 2012 Kolchin s generalized allocation scheme A law of

### STAT 3502. x 0 < x < 1

Solution - Assignment # STAT 350 Total mark=100 1. A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous

### Binomial random variables

Binomial and Poisson Random Variables Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance

### Exploratory Data Analysis

Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

### The Binomial Distribution. Summer 2003

The Binomial Distribution Summer 2003 Internet Bubble Several industry experts believe that 30% of internet companies will run out of cash in 6 months and that these companies will find it very hard to

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions

Math 370, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

### Sums of Independent Random Variables

Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables

### Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

### LECTURE 16. Readings: Section 5.1. Lecture outline. Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process

LECTURE 16 Readings: Section 5.1 Lecture outline Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process Number of successes Distribution of interarrival times The

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

### 3. The Economics of Insurance

3. The Economics of Insurance Insurance is designed to protect against serious financial reversals that result from random evens intruding on the plan of individuals. Limitations on Insurance Protection

### MAT 155. Key Concept. September 27, 2010. 155S5.5_3 Poisson Probability Distributions. Chapter 5 Probability Distributions

MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance and Standard

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.

### Chapter 5: Normal Probability Distributions - Solutions

Chapter 5: Normal Probability Distributions - Solutions Note: All areas and z-scores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that

### Financial Mathematics and Simulation MATH 6740 1 Spring 2011 Homework 2

Financial Mathematics and Simulation MATH 6740 1 Spring 2011 Homework 2 Due Date: Friday, March 11 at 5:00 PM This homework has 170 points plus 20 bonus points available but, as always, homeworks are graded

### 39.2. The Normal Approximation to the Binomial Distribution. Introduction. Prerequisites. Learning Outcomes

The Normal Approximation to the Binomial Distribution 39.2 Introduction We have already seen that the Poisson distribution can be used to approximate the binomial distribution for large values of n and

### LECTURE - 1 INTRODUCTION TO QUEUING SYSTEM

LECTURE - 1 INTRODUCTION TO QUEUING SYSTEM Learning objective To introduce features of queuing system 9.1 Queue or Waiting lines Customers waiting to get service from server are represented by queue and

EXAM 3, FALL 003 Please note: On a one-time basis, the CAS is releasing annotated solutions to Fall 003 Examination 3 as a study aid to candidates. It is anticipated that for future sittings, only the

### Chapter 4. Probability Distributions

Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive

### Binomial random variables (Review)

Poisson / Empirical Rule Approximations / Hypergeometric Solutions STAT-UB.3 Statistics for Business Control and Regression Models Binomial random variables (Review. Suppose that you are rolling a die

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Generating Functions

Chapter 10 Generating Functions 10.1 Generating Functions for Discrete Distributions So far we have considered in detail only the two most important attributes of a random variable, namely, the mean and

### Lecture 13: Martingales

Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

### 6 POISSON DISTRIBUTIONS

6 POISSON DISTRIBUTIONS Chapter 6 Poisson Distributions Objectives After studying this chapter you should be able to recognise when to use the Poisson distribution; be able to apply the Poisson distribution

### 7: The CRR Market Model

Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein

### SIMULATING CANCELLATIONS AND OVERBOOKING IN YIELD MANAGEMENT

CHAPTER 8 SIMULATING CANCELLATIONS AND OVERBOOKING IN YIELD MANAGEMENT In YM, one of the major problems in maximizing revenue is the number of cancellations. In industries implementing YM this is very

### ECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003

ECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003 1. (4 points) The number of claims for missing baggage for a well-known airline

### CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

### Resource Provisioning and Network Traffic

Resource Provisioning and Network Traffic Network engineering: Feedback traffic control closed-loop control ( adaptive ) small time scale: msec mainly by end systems e.g., congestion control Resource provisioning

### What is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference

0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures

### Bayesian logistic betting strategy against probability forecasting. Akimichi Takemura, Univ. Tokyo. November 12, 2012

Bayesian logistic betting strategy against probability forecasting Akimichi Takemura, Univ. Tokyo (joint with Masayuki Kumon, Jing Li and Kei Takeuchi) November 12, 2012 arxiv:1204.3496. To appear in Stochastic

### Summer School in Statistics for Astronomers & Physicists, IV June 9-14, 2008

p. 1/4 Summer School in Statistics for Astronomers & Physicists, IV June 9-14, 2008 Laws of Probability, Bayes theorem, and the Central Limit Theorem June 10, 8:45-10:15 am Mosuk Chow Department of Statistics

### 4 The M/M/1 queue. 4.1 Time-dependent behaviour

4 The M/M/1 queue In this chapter we will analyze the model with exponential interarrival times with mean 1/λ, exponential service times with mean 1/µ and a single server. Customers are served in order

### Lectures on Stochastic Processes. William G. Faris

Lectures on Stochastic Processes William G. Faris November 8, 2001 2 Contents 1 Random walk 7 1.1 Symmetric simple random walk................... 7 1.2 Simple random walk......................... 9 1.3

### FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### Statistics 104: Section 6!

Page 1 Statistics 104: Section 6! TF: Deirdre (say: Dear-dra) Bloome Email: dbloome@fas.harvard.edu Section Times Thursday 2pm-3pm in SC 109, Thursday 5pm-6pm in SC 705 Office Hours: Thursday 6pm-7pm SC

### 1 The Brownian bridge construction

The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof

### Mathematical Expectation

Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the

### SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

### Random Variate Generation (Part 3)

Random Variate Generation (Part 3) Dr.Çağatay ÜNDEĞER Öğretim Görevlisi Bilkent Üniversitesi Bilgisayar Mühendisliği Bölümü &... e-mail : cagatay@undeger.com cagatay@cs.bilkent.edu.tr Bilgisayar Mühendisliği

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA

VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA Csilla Csendes University of Miskolc, Hungary Department of Applied Mathematics ICAM 2010 Probability density functions A random variable X has density

### The Math. P (x) = 5! = 1 2 3 4 5 = 120.

The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct

### 0 x = 0.30 x = 1.10 x = 3.05 x = 4.15 x = 6 0.4 x = 12. f(x) =

. A mail-order computer business has si telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. 0 2 3 4 5 6 p(.0.5.20.25.20.06.04

### Notes from Week 1: Algorithms for sequential prediction

CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking

### MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

### Statistics 100A Homework 4 Solutions

Chapter 4 Statistics 00A Homework 4 Solutions Ryan Rosario 39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn.

### Lecture Notes 1. Brief Review of Basic Probability

Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters -3 are a review. I will assume you have read and understood Chapters -3. Here is a very

### SOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION. School of Mathematical Sciences. Monash University, Clayton, Victoria, Australia 3168

SOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION Ravi PHATARFOD School of Mathematical Sciences Monash University, Clayton, Victoria, Australia 3168 In this paper we consider the problem of gambling with

### Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

### Traffic Behavior Analysis with Poisson Sampling on High-speed Network 1

Traffic Behavior Analysis with Poisson Sampling on High-speed etwork Guang Cheng Jian Gong (Computer Department of Southeast University anjing 0096, P.R.China) Abstract: With the subsequent increasing

### Martingale Ideas in Elementary Probability. Lecture course Higher Mathematics College, Independent University of Moscow Spring 1996

Martingale Ideas in Elementary Probability Lecture course Higher Mathematics College, Independent University of Moscow Spring 1996 William Faris University of Arizona Fulbright Lecturer, IUM, 1995 1996

### Review #2. Statistics

Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2

Math 370, Spring 2008 Prof. A.J. Hildebrand Practice Test 2 About this test. This is a practice test made up of a random collection of 15 problems from past Course 1/P actuarial exams. Most of the problems

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

### MAINTAINED SYSTEMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University ENGINEERING RELIABILITY INTRODUCTION

MAINTAINED SYSTEMS Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University OUTLINE MAINTE MAINTE MAINTAINED UNITS Maintenance can be employed in two different manners: Preventive

### Introduction to Probability

Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

### A LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA

REVSTAT Statistical Journal Volume 4, Number 2, June 2006, 131 142 A LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA Authors: Daiane Aparecida Zuanetti Departamento de Estatística, Universidade Federal de São

### STAT 360 Probability and Statistics. Fall 2012

STAT 360 Probability and Statistics Fall 2012 1) General information: Crosslisted course offered as STAT 360, MATH 360 Semester: Fall 2012, Aug 20--Dec 07 Course name: Probability and Statistics Number

### 1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

### Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

### Future Value of an Annuity Sinking Fund. MATH 1003 Calculus and Linear Algebra (Lecture 3)

MATH 1003 Calculus and Linear Algebra (Lecture 3) Future Value of an Annuity Definition An annuity is a sequence of equal periodic payments. We call it an ordinary annuity if the payments are made at the

### The Variability of P-Values. Summary

The Variability of P-Values Dennis D. Boos Department of Statistics North Carolina State University Raleigh, NC 27695-8203 boos@stat.ncsu.edu August 15, 2009 NC State Statistics Departement Tech Report

### Examples: Joint Densities and Joint Mass Functions Example 1: X and Y are jointly continuous with joint pdf

AMS 3 Joe Mitchell Eamples: Joint Densities and Joint Mass Functions Eample : X and Y are jointl continuous with joint pdf f(,) { c 2 + 3 if, 2, otherwise. (a). Find c. (b). Find P(X + Y ). (c). Find marginal

### Chapter 4. iclicker Question 4.4 Pre-lecture. Part 2. Binomial Distribution. J.C. Wang. iclicker Question 4.4 Pre-lecture

Chapter 4 Part 2. Binomial Distribution J.C. Wang iclicker Question 4.4 Pre-lecture iclicker Question 4.4 Pre-lecture Outline Computing Binomial Probabilities Properties of a Binomial Distribution Computing

### M/M/1 and M/M/m Queueing Systems

M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General - can be any distribution. First letter: Arrival process; M: memoryless - exponential

### Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem