Questions and Answers

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Questions and Answers"

Transcription

1 GNH7/GEOLGG9/GEOL2 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD TUTORIAL (6): EARTHQUAKE STATISTICS Question. Questions and Answers How many distinct 5-card hands can be dealt from a standard 52-card deck? Answer. Since the 5-card hand remains unchanged if you received the same 5 cards, but in a different order, the answer is, 52 C 5 = 52! / (7! 5!) = 2,598,96. Question 2. It is frequency assumed phenomena such as injury-causing accidents in a large industrial plant satisfy the three assumptions for a Poisson process. Suppose these assumptions are true for a particular plant, occurring at a rate of say, λ = ½ per week. What is the probability that eactly N = number of accidents will occur in the net 6 weeks? What is the mean number of accidents that should occur in this time and what is the variance? What is the probability of at least accidents? What is the probability of no accidents? Plot the histograms for µ= and µ=8. Answer 2 If N represents the number of such earthquakes to occur in the net t = 6 weeks, then N is a Poisson random variable with parameter µ = ½(6) = and its probability function is p N ( ) = e!, =,,2,... Both the mean and variance for N equal. The probability of eactly earthquakes in this period is p N () = e! =.22 And the probability of at least earthquakes is

2 = e! =.5 The probability there will be no earthquakes is p N () = e! =.5 Sketch the histograms with µ= and µ=8. Mark on the mean and standard deviation...5. p() Question Plot the cumulative distribution function for the eample. Answer.75 F(t) t 2

3 Question Assuming that destructive earthquakes are occurring in a region at a rate of λ = ½ per decade, we will try and predict earthquakes occurrence from now, year 2, what is the epected number of years and the standard deviation for the first earthquake to occur? What is the probability that the first decade will be earthquake free? What is the probability that the first earthquake will occur in 25, and that it occurs in 28? Answer Given that destructive earthquakes are occurring in a region at a rate of λ = ½ per decade, this is equivalent to occurrences at the rate of /2 per year. We can predict the number of earthquakes we could epect to occur from now. If T is the number of years until the first earthquake occurs, then T is eponential with λ = /2. The epected number of years to the first earthquake is E[T] = (/2) - = 2 and the standard deviation for T is σ T = 2 years The probability that the first decade is earthquake free is: Pr[T>] = e - ()/2 =.67 The probability that the first earthquake occurs in 25 is Pr[ < T 5] = ( e - (5)/2 ) ( e - ()/2 ) =. and the probability that it occurs in 28 is Pr[7 < T 8] = ( e -(8)/2 ( e -(7)/2 ) =.5 Question 5 Show that the time interval distribution function, F T (), for slip-predictable behaviour in the Shimazaki model, with constant strain rate, k, is given by: F T () = ep[-(k/2). 2 ] Hint: Solution to the partial differential equation df()/d = - k.t.f() is f() = f().ep[(-k/2). 2 ].

4 Answer 5 We can t make assumptions about Poisson statistics and have to derive this from first principals. That is the whole point. In both the Shimazaki slip- and time-predictable models the strain rate is constant, k. For the slip-predictable model the probability of rupture is proportional to the accumulated strain, kt, where t is time. Given that no earthquake has occurred, the probability of an earthquake occurring in a small time interval t at time t is: We now consider time intervals, and to avoid confusion we change notation to. The probability that an earthquake occurs at any time,, we will denote P T (). The probability the interval T to the net earthquake will be greater than is the probability that the earthquake will occur in time + : Substituting: Pr[ N( t, t + t) = ] = kt t Pr[T > + ] = P T ( + ) = P T (). Pr[N(, + ) =] P T ( + ) = P T ().( k ) Now, So, P T ( + ) P T () = dp T (): dp T () = - k P T () d The solution of this partial differential equation is P T = P T () ep(-k 2 /2) or, since P T () =, F T () = ep(-ξ 2 ) where ξ = k/2. This is a Weibull distribution with eponent n = 2. So the Shimazaki model leads to a Weibull distribution. To get the probability density function we differentiate.

5 Supplementary Questions Question. Let us assume that the number of telephone calls to UCL are made in accordance with a Poisson process assumptions at the rate of 2 per hour during the period 9. to 2.. What is the epected number of calls during this period and the epected number in any single second? What is the probability of at least one call being made in any given minute and the probability of at least one call being made during any given second? Question 2. What is the probability of an earthquake occurring for the Shimazaki time-predictable model? What is the probability density function? 5

12.5: CHI-SQUARE GOODNESS OF FIT TESTS

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

More information

Chapter 5 Discrete Probability Distribution. Learning objectives

Chapter 5 Discrete Probability Distribution. Learning objectives Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able

More information

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part 1

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part 1 Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment Statistics and Probability Example Part Note: Assume missing data (if any) and mention the same. Q. Suppose X has a normal

More information

Lecture 8: More Continuous Random Variables

Lecture 8: More Continuous Random Variables Lecture 8: More Continuous Random Variables 26 September 2005 Last time: the eponential. Going from saying the density e λ, to f() λe λ, to the CDF F () e λ. Pictures of the pdf and CDF. Today: the Gaussian

More information

2 Binomial, Poisson, Normal Distribution

2 Binomial, Poisson, Normal Distribution 2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability

More information

DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced

More information

9-3.4 Likelihood ratio test. Neyman-Pearson lemma

9-3.4 Likelihood ratio test. Neyman-Pearson lemma 9-3.4 Likelihood ratio test Neyman-Pearson lemma 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental

More information

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is. Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

More information

the number of organisms in the squares of a haemocytometer? the number of goals scored by a football team in a match?

the number of organisms in the squares of a haemocytometer? the number of goals scored by a football team in a match? Poisson Random Variables (Rees: 6.8 6.14) Examples: What is the distribution of: the number of organisms in the squares of a haemocytometer? the number of hits on a web site in one hour? the number of

More information

Math 431 An Introduction to Probability. Final Exam Solutions

Math 431 An Introduction to Probability. Final Exam Solutions Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

More information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

UNIVERSITY OF OSLO. The Poisson model is a common model for claim frequency.

UNIVERSITY OF OSLO. The Poisson model is a common model for claim frequency. UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Candidate no Exam in: STK 4540 Non-Life Insurance Mathematics Day of examination: December, 9th, 2015 Examination hours: 09:00 13:00 This

More information

Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams

Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Review for Final Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Histogram Boxplots Chapter 3: Set

More information

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8. Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

More information

The Normal distribution

The Normal distribution The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

More information

Lecture 5 : The Poisson Distribution

Lecture 5 : The Poisson Distribution Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

More information

Joint Exam 1/P Sample Exam 1

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

More information

Worked examples Multiple Random Variables

Worked examples Multiple Random Variables Worked eamples Multiple Random Variables Eample Let X and Y be random variables that take on values from the set,, } (a) Find a joint probability mass assignment for which X and Y are independent, and

More information

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course On STRUCTURAL RELIABILITY Module # 02 Lecture 6 Course Format: Web Instructor: Dr. Arunasis Chakraborty Department of Civil Engineering Indian Institute of Technology Guwahati 6. Lecture 06:

More information

Exploratory Data Analysis

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

More information

Chapter 9 Monté Carlo Simulation

Chapter 9 Monté Carlo Simulation MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?

More information

The Binomial Probability Distribution

The Binomial Probability Distribution The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

More information

Chapter 4. Probability Distributions

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

More information

Probability Calculator

Probability Calculator Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that

More information

MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

More information

Normal Distribution. Definition A continuous random variable has a normal distribution if its probability density. f ( y ) = 1.

Normal Distribution. Definition A continuous random variable has a normal distribution if its probability density. f ( y ) = 1. Normal Distribution Definition A continuous random variable has a normal distribution if its probability density e -(y -µ Y ) 2 2 / 2 σ function can be written as for < y < as Y f ( y ) = 1 σ Y 2 π Notation:

More information

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

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

More information

Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22

Math 151. Rumbos Spring 2014 1. Solutions to Assignment #22 Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability

More information

Binomial Random Variables

Binomial Random Variables Binomial Random Variables Dr Tom Ilvento Department of Food and Resource Economics Overview A special case of a Discrete Random Variable is the Binomial This happens when the result of the eperiment is

More information

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

More information

Statistics 100A Homework 4 Solutions

Statistics 100A Homework 4 Solutions Problem 1 For a discrete random variable X, Statistics 100A Homework 4 Solutions Ryan Rosario Note that all of the problems below as you to prove the statement. We are proving the properties of epectation

More information

Introduction to Basic Reliability Statistics. James Wheeler, CMRP

Introduction to Basic Reliability Statistics. James Wheeler, CMRP James Wheeler, CMRP Objectives Introduction to Basic Reliability Statistics Arithmetic Mean Standard Deviation Correlation Coefficient Estimating MTBF Type I Censoring Type II Censoring Eponential Distribution

More information

Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

More information

Chapter 5: Normal Probability Distributions - Solutions

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

More information

Marginal Functions in Economics

Marginal Functions in Economics Marginal Functions in Economics One of the applications of derivatives in a real world situation is in the area of marginal analysis. Marginal analysis uses the derivative (or rate of change) to determine

More information

Beta Distribution. Paul Johnson <pauljohn@ku.edu> and Matt Beverlin <mbeverlin@ku.edu> June 10, 2013

Beta Distribution. Paul Johnson <pauljohn@ku.edu> and Matt Beverlin <mbeverlin@ku.edu> June 10, 2013 Beta Distribution Paul Johnson and Matt Beverlin June 10, 2013 1 Description How likely is it that the Communist Party will win the net elections in Russia? In my view,

More information

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...

Hypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test... Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................

More information

3.4 The Normal Distribution

3.4 The Normal Distribution 3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous

More information

Key Concept. Density Curve

Key Concept. Density Curve MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 6 Normal Probability Distributions 6 1 Review and Preview 6 2 The Standard Normal Distribution 6 3 Applications of Normal

More information

Notes on the Negative Binomial Distribution

Notes on the Negative Binomial Distribution Notes on the Negative Binomial Distribution John D. Cook October 28, 2009 Abstract These notes give several properties of the negative binomial distribution. 1. Parameterizations 2. The connection between

More information

Notes on Continuous Random Variables

Notes on Continuous Random Variables Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

More information

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

More information

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit Chi-Square Tests 15 Chapter Chi-Square Test for Independence Chi-Square Tests for Goodness Uniform Goodness- Poisson Goodness- Goodness Test ECDF Tests (Optional) McGraw-Hill/Irwin Copyright 2009 by The

More information

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

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

More information

How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

More information

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures. Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

More information

Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model

Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model These notes consider the single-period model in Kyle (1985) Continuous Auctions and Insider Trading, Econometrica 15,

More information

Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:

Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve

More information

Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2. Hypothesis testing Power Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

More information

STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)

STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I) The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable

More information

2. DATA AND EXERCISES (Geos2911 students please read page 8)

2. DATA AND EXERCISES (Geos2911 students please read page 8) 2. DATA AND EXERCISES (Geos2911 students please read page 8) 2.1 Data set The data set available to you is an Excel spreadsheet file called cyclones.xls. The file consists of 3 sheets. Only the third is

More information

6.2 Normal distribution. Standard Normal Distribution:

6.2 Normal distribution. Standard Normal Distribution: 6.2 Normal distribution Slide Heights of Adult Men and Women Slide 2 Area= Mean = µ Standard Deviation = σ Donation: X ~ N(µ,σ 2 ) Standard Normal Distribution: Slide 3 Slide 4 a normal probability distribution

More information

Mark Scheme 4767 June 2005 GENERAL INSTRUCTIONS Marks in the mark scheme are explicitly designated as M, A, B, E or G. M marks ("method") are for an attempt to use a correct method (not merely for stating

More information

WHERE DOES THE 10% CONDITION COME FROM?

WHERE DOES THE 10% CONDITION COME FROM? 1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

More information

CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS

CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS CHAPTER 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS CENTRAL LIMIT THEOREM (SECTION 7.2 OF UNDERSTANDABLE STATISTICS) The Central Limit Theorem says that if x is a random variable with any distribution having

More information

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

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

More information

Binomial random variables

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

More information

Practice problems for Homework 11 - Point Estimation

Practice problems for Homework 11 - Point Estimation Practice problems for Homework 11 - Point Estimation 1. (10 marks) Suppose we want to select a random sample of size 5 from the current CS 3341 students. Which of the following strategies is the best:

More information

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

More information

2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table 2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

More information

MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample

MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample MATH 10: Elementary Statistics and Probability Chapter 9: Hypothesis Testing with One Sample Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of

More information

7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

More information

Binomial random variables (Review)

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

More information

P(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b

P(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b Continuous Random Variables The probability that a continuous random variable, X, has a value between a and b is computed by integrating its probability density function (p.d.f.) over the interval [a,b]:

More information

The Standard Normal distribution

The Standard Normal distribution The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance

More information

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. 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...................................

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

More information

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

More information

Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

More information

MATH 140 Lab 4: Probability and the Standard Normal Distribution

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

More information

Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8

More information

1. A survey of a group s viewing habits over the last year revealed the following

1. A survey of a group s viewing habits over the last year revealed the following 1. A survey of a group s viewing habits over the last year revealed the following information: (i) 8% watched gymnastics (ii) 9% watched baseball (iii) 19% watched soccer (iv) 14% watched gymnastics and

More information

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

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

More information

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 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

More information

Normal distribution. ) 2 /2σ. 2π σ

Normal distribution. ) 2 /2σ. 2π σ Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

More information

Solutions to Worksheet on Hypothesis Tests

Solutions to Worksheet on Hypothesis Tests s to Worksheet on Hypothesis Tests. A production line produces rulers that are supposed to be inches long. A sample of 49 of the rulers had a mean of. and a standard deviation of.5 inches. The quality

More information

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to, LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

More information

1 Sufficient statistics

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 =

More information

MAS113 Introduction to Probability and Statistics Exercises

MAS113 Introduction to Probability and Statistics Exercises MAS113 Introduction to Probability and Statistics Exercises You will be told which problems to work on each week at the lecture each Monday, and you should try them before your class each Wednesday. Every

More information

Confidence Intervals for the Difference Between Two Means

Confidence Intervals for the Difference Between Two Means Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

More information

CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS

CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS TRUE/FALSE 235. The Poisson probability distribution is a continuous probability distribution. F 236. In a Poisson distribution,

More information

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1]. Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

More information

6. Jointly Distributed Random Variables

6. Jointly Distributed Random Variables 6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.

More information

, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0

, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0 Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after Simeon-Denis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will

More information

Algebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED

Algebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED Algebra Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED. Graph eponential functions. (Sections 7., 7.) Worksheet 6. Solve eponential growth and eponential decay problems. (Sections 7., 7.) Worksheet 8.

More information

Chapter 6 MORE ABOUT RANDOM VARIABLES. Mean and variance of a random variable...

Chapter 6 MORE ABOUT RANDOM VARIABLES. Mean and variance of a random variable... This work is licensed under the Creative commons Attribution-Non-commercial-Share Alike 2.5 South Africa License. You are free to copy, communicate and adapt the work on condition that you attribute the

More information

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions... MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

More information

INSURANCE RISK THEORY (Problems)

INSURANCE RISK THEORY (Problems) INSURANCE RISK THEORY (Problems) 1 Counting random variables 1. (Lack of memory property) Let X be a geometric distributed random variable with parameter p (, 1), (X Ge (p)). Show that for all n, m =,

More information

SECTION 1-4 Absolute Value in Equations and Inequalities

SECTION 1-4 Absolute Value in Equations and Inequalities 1-4 Absolute Value in Equations and Inequalities 37 SECTION 1-4 Absolute Value in Equations and Inequalities Absolute Value and Distance Absolute Value in Equations and Inequalities Absolute Value and

More information

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm

99.37, 99.38, 99.38, 99.39, 99.39, 99.39, 99.39, 99.40, 99.41, 99.42 cm Error Analysis and the Gaussian Distribution In experimental science theory lives or dies based on the results of experimental evidence and thus the analysis of this evidence is a critical part of the

More information

Lesson 20. Probability and Cumulative Distribution Functions

Lesson 20. Probability and Cumulative Distribution Functions Lesson 20 Probability and Cumulative Distribution Functions Recall If p(x) is a density function for some characteristic of a population, then Recall If p(x) is a density function for some characteristic

More information

AP STATISTICS 2010 SCORING GUIDELINES

AP STATISTICS 2010 SCORING GUIDELINES 2010 SCORING GUIDELINES Question 4 Intent of Question The primary goals of this question were to (1) assess students ability to calculate an expected value and a standard deviation; (2) recognize the applicability

More information

An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

More information

Capital Market Theory: An Overview. Return Measures

Capital Market Theory: An Overview. Return Measures Capital Market Theory: An Overview (Text reference: Chapter 9) Topics return measures measuring index returns (not in text) holding period returns return statistics risk statistics AFM 271 - Capital Market

More information

MATHEMATICS Extended Part Module 1 (Calculus and Statistics) (Sample Paper)

MATHEMATICS Extended Part Module 1 (Calculus and Statistics) (Sample Paper) HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION MATHEMATICS Extended Part Module 1 (Calculus and Statistics) (Sample Paper) Time allowed: hours 30 minutes

More information

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

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

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

2WB05 Simulation Lecture 8: Generating random variables

2WB05 Simulation Lecture 8: Generating random variables 2WB05 Simulation Lecture 8: Generating random variables Marko Boon http://www.win.tue.nl/courses/2wb05 January 7, 2013 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions Generating

More information