Reading Material for Students
|
|
- Hilary Glenn
- 5 years ago
- Views:
Transcription
1 Reading Material for Students Arnab Adhikari Indian Institute of Management Calcutta, Joka, Kolkata 714, India, Indranil Biswas Indian Institute of Management Lucknow, Prabandh Nagar, Lucknow 613, India, Arnab Bisi Johns opkins Carey Business School, 1 International Drive, Baltimore, Maryland 1, abisi1@jhu.edu Probability Distributions Binomial distribution. The Binomial distribution describes the probability of exactly x successes out of N trials; the probability associated with a success in a single trial is given by p and that with a failure is given by 1 p (also designated by q). The expression of the probability mass function (pmf) of this distribution is as follows p(x; N, p) = ( N x )px (1 p) N x, where the variable x and the parameter N are integers, satisfying the conditions x N and N >. The parameter p is a real quantity and p [,1]. The expected value and the variance of a random variable X having binomial distribution can be expressed as follows: and Var X N p (1 p ) X N p. ypergeometric distribution. The hypergeometric distribution describes the experiment where out of total N elements, M possesses a certain attribute [and the remaining (N M) does not]; if we then choose n elements at random without replacement, p(x; n, N, M) gives the probability that exactly x of the selected n elements have come from the group of M elements that possesses the attribute. Let the number of elements with that certain attribute be denoted by X. The probability mass function (pmf) of X with hypergeometric distribution is given by f(x; n, N, M) = (M x )(N M n x ) ( N n ) where x is discrete and its range is given by: x [max(, n N + M), min (n, M)]. The parameters n, N and M are all integers and satisfy the following conditions: 1 n N, N 1 1
2 and M 1. Let probability of success be represented by M p N the variance of X under hypergeometric distribution can be expressed as follows: X np and X np (1 p) ( N n) Var. ( N 1). Then, the expected value and In real life, when a marketing group is trying to understand their customer base by testing a set of known customers for over-representation of various demographic subgroups, they use hypergeometric test designed based on hypergeometric distribution. Negative Binomial distribution. The negative binomial distribution (also known as Pascal distribution) gives the probability of waiting for exactly x trials until k th success has occurred. Let the number of trials before k th success be denoted by X. ere p and q(= 1 p) designates the probability of a success and a failure in a single trial, respectively. The probability mass function (pmf) of this distribution is given by f(x; k, p) = ( x 1 k 1 )pk (1 p) x k, where the variable x and parameter k are integers and satisfies the following condition: x k >. Now, the expected value and the variance of a random variable X under negative binomial distribution can be expressed as follows: X k ( 1 and Var X p p) k (1 p). p The negative binomial distribution has applications in the insurance industry, where for example the rate at which people have accidents is affected by a random variable like the weather condition. Geometric distribution. The geometric distribution is a special case of the negative binomial distribution discussed above with k = 1. It expresses the probability of waiting for exactly x trials before the occurrence of the first successful event. Let the number of trials before the first success be denoted by X. Then, the probability mass function (pmf) of X with this distribution is given by f(x; p) = p(1 p) x 1, where p denotes the probability of success in each trial. The expected value and the variance of a random variable X under geometric distribution can be expressed as follows:
3 X ( 1 and Var X p p) (1 p). p In real life, if a NGO wants to know the number of male births before one female birth regarding the study of sex ratio in human population then it can use this kind of distribution. Poisson distribution. The Poisson distribution gives the probability of finding occurrence of exactly x events in a given length of time when the events are independent in nature and happens at a constant rate, given by. The probability mass function (pmf) of this distribution is given by e f ( x ; ), x! where the variable x is a positive integer and the parameter is a real positive quantity. Now, the expected value and the variance of a random variable X under Poisson distribution can be expressed as follows: X and X x Var. When the value of N is very large and p is very small in the binomial distribution described before, then it can be approximated by a Poisson distribution with expected value = Np. Poisson distribution is applied to determine the probability of rare events like birth defects, genetic mutations, car accidents, etc. Uniform distribution. If a continuous random variable X follows the uniform distribution, then its probability density function (pdf) is given by the expression f(x; a, b) = 1 b a for a X b. The expected value and the variance of a random variable X under uniform distribution can be expressed as follows X b a and Var X b a. 1 In oil exploration, the position of the oil-water contact in a potential prospect is often considered to be uniformly distributed. xponential distribution. If a continuous random variable X follows the exponential distribution, then its pdf can be expressed as follows: 3
4 f(x; θ) = 1 θ e x θ, where θ represents the scale parameter. The expected value and the variance of a random variable X under exponential distribution are given by: X and X Var. In real life, the radioactive or particle decays is considered to follow exponential distribution. Normal distribution. The normal distribution (also called the Gauss distribution) is one of the most important distributions in statistics. The pdf of normal distribution is given by the following expression: f(x; μ, σ ) = 1 1 σ π e (x μ σ ), where μ is the mean or expected value and σ is the variance of the distribution. For μ = and σ = 1, the distribution is called the standard normal distribution. It has widespread applications in natural and social sciences, financial models, etc. Beta distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The pdf of beta distribution is given by: f(x; α, β) = 1 B(α,β) xα 1 (1 x) β 1, where the shape parameters α and β are positive real numbers, and the variable x satisfies the condition x 1. B(α, β) designates the beta function and is given by the following expression B(α, β) = Γ(α)Γ(β) Γ(α+β). For α R +, the gamma function Γ(α) is defined by the integral Γ(α) = t α 1 e t dt. When α = β = 1, the beta distribution assumes the form of the uniform distribution between and 1; when α = β = the distribution takes parabolic shape; when α = and β = 1 or vise versa the distribution takes triangular shaped distribution. The expected value and the variance of a random variable X under beta distribution can be expressed as follows: X and X Var. ( 1) 4
5 Beta distribution is usually applied to determine the time allocation in project management/ control systems, heterogeneity in the probability of IV transmission, etc. Gamma distribution. It is a two-parameter family of continuous probability distributions. xponential distribution is a special case of the gamma distribution. The pdf of gamma distribution can be represented by the following functional form: f(x; k, θ) = xk 1 e x θ, θ k Γ(k) where the shape parameter k and the scale parameter θ are positive real numbers (k R + and θ R + ) and the variable x is also a positive real number (x R + ). The expected value and the variance of a random variable X under gamma distribution are given by: X k and X Var k. Sampling Distribution and Confidence Interval. If we take repeated samples from the same population, samples means x would vary from sample to sample and form a sampling distribution of sample means. It explains the random behavior of a sample mean. The variability of x from can be obtained by determining the variance of x. The variance of the sample mean with a sample of size n is given by:. x n Next, the confidence interval contains the true population parameter. A confidence interval comprise point estimate, i.e., the best estimate of the population parameter from the sample statistic and the margin of error or maximum sampling error (the maximum accepted difference between the true population parameter and a sample estimate of that parameter). The confidence interval where lies can be determined by the following expression: x z x z / / n n The confidence level is denoted by 1 1 %. The margin of error denoted by is given by the following formula:. z / n. 5
6 From the formula given above, the required minimum sample size can be easily obtained and it is given by: n z ( / ). ypothesis Testing. ypothesis testing is a technique to check with the help of a sample data whether a claim or hypothesis about a population parameter is true or not. In hypothesis testing, the stated conjecture defined as the null hypothesis can be disproved, but it cannot be proved. owever, by disproving the null hypothesis, one can prove that the contrary is true. The contrary of the null hypothesis is termed as the alternative hypothesis. The test statistic represents the value determined using the sample data. A test statistic for testing a hypothesis on population mean is given by the following formula: z x, n where denotes the hypothesized value of the population mean. Following are the null ( ) and alternative ( The Two-Tailed Test. A ) hypotheses for three standard tests on population mean: : A z : x n reject if z z or z z. / / The One-Tailed Test to the Right : A z : x n reject if. z z 6
7 The One-Tailed Test to the Left : A : x z n reject if. z z Regression Models Simple linear regression ere we present a simple linear regression model to determine the relationship between the dependent variable Y and the independent variable X, captured by the following equation: ( Y X) = α + βx. Then the regression model can be designated as: Y = α + βx + ε, where ε = Y ( Y X) is a random variable or an error term with (ε) = and Var ( ε) = σ. If α and β denote the best estimates of the parameters α and β, respectively, then the estimated linear regression equation of Y on X is: Multiple linear regression Y = α + β X. The effect of independent variables X 1, X and X 3 on the dependent variable Y can be captured by the following equation: ( Y X 1, X, X 3 ) = α + β 1 X 1 + β X + β 3 X 3, where ε = Y ( Y X 1, X, X 3 ) is a random variable or an error term with (ε) = and Var( ε) = σ. If α, β, 1 β, and β 3 denote the best estimates of the parameters α, β 1, β, and β 3, respectively, then the estimated multiple linear regression equation of Y on X 1, X and X 3 is given by: Multicollinearity check Y = α + β X β X + β X 3 3. Often regression model is affected by linear relationship between independent variables termed as multicollinearity. Variance Inflation Factor (VIF) is one of the conventional techniques employed to check whether any multicollinearity exists or not. VIF between two independent variables X 1 and X can be determined by the following expression: 7
8 VIF X1,X = 1 1 R X 1,X, where R X1,X denotes the co-efficient of determination between X 1 and X. If the value of VIF is greater than 5, then it indicates multicollinearity and the overall regression model gets affected by it. Sources Anderson, D., Sweeney, D., Williams, T., Camm, J., Cochran, J. 11. Statistics for Business & conomics, 11 th ed. Cengage Learning, Mason. Berenson, M., Levine, D., Krehbiel, T. C. 11. Basic business statistics: Concepts and applications. Pearson ducation, New Jersey. Groebner, D. F., Shannon, P. W., Fry, P. C., Smith, K. D. 13. Business statistics: a decision making approach, 9 th ed. Pearson ducation, New Jersey. ildebrand, D. K. and O. Lyman Statistical Thinking for Managers, 4 th ed. Duxbury Press, California. Levin, R. I. and D. S. Rubin Statistics for Management, 7 th ed. Prentice all International, New Jersey
Brief Review of Probability
Maura Department of Economics and Finance Università Tor Vergata Outline 1 Distribution Functions Quantiles and Modes of a Distribution 2 Example 3 Example 4 Distributions Outline Distribution Functions
More informationBinomial and Poisson Probability Distributions
Binomial and Poisson Probability Distributions Esra Akdeniz March 3, 2016 Bernoulli Random Variable Any random variable whose only possible values are 0 or 1 is called a Bernoulli random variable. What
More informationGEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs
STATISTICS 4 Summary Notes. Geometric and Exponential Distributions GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs P(X = x) = ( p) x p x =,, 3,...
More informationTUTORIAL 8 SOLUTIONS #
TUTORIAL 8 SOLUTIONS #9.11.21 Suppose that a single observation X is taken from a uniform density on [0,θ], and consider testing H 0 : θ = 1 versus H 1 : θ =2. (a) Find a test that has significance level
More informationProbability Distributions Columns (a) through (d)
Discrete Probability Distributions Columns (a) through (d) Probability Mass Distribution Description Notes Notation or Density Function --------------------(PMF or PDF)-------------------- (a) (b) (c)
More informationThe University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80
The University of Hong Kong Department of Statistics and Actuarial Science STAT2802 Statistical Models Tutorial Solutions Solutions to Problems 71-80 71. Decide in each case whether the hypothesis is simple
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 informationThings to remember when learning probability distributions:
SPECIAL DISTRIBUTIONS Some distributions are special because they are useful They include: Poisson, exponential, Normal (Gaussian), Gamma, geometric, negative binomial, Binomial and hypergeometric distributions
More information3 Modeling Process Quality
3 Modeling Process Quality 3.1 Introduction Section 3.1 contains basic numerical and graphical methods. familiar with these methods. It is assumed the student is Goal: Review several discrete and continuous
More informationPARAMETER ESTIMATION: BAYESIAN APPROACH. These notes summarize the lectures on Bayesian parameter estimation.
PARAMETER ESTIMATION: BAYESIAN APPROACH. These notes summarize the lectures on Bayesian parameter estimation.. Beta Distribution We ll start by learning about the Beta distribution, since we end up using
More informationSolutions to the Spring 2015 CAS Exam ST
Solutions to the Spring 2015 CAS Exam ST (updated to include the CAS Final Answer Key of July 15) There were 25 questions in total, of equal value, on this 2.5 hour exam. There was a 10 minute reading
More informationContinuous Random Variables
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random variables
More information2.3 Analysis of Categorical Data
90 CHAPTER 2. ESTIMATION AND HYPOTHESIS TESTING 2.3 Analysis of Categorical Data 2.3.1 The Multinomial Probability Distribution A mulinomial random variable is a generalization of the binomial rv. It results
More informationRandom Variables. Definition: A random variable (r.v.) X on the probability space (Ω, F, P) is a mapping
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More informationChapter 1. Sets and probability. 1.3 Probability space
Random processes - Chapter 1. Sets and probability 1 Random processes Chapter 1. Sets and probability 1.3 Probability space 1.3 Probability space Random processes - Chapter 1. Sets and probability 2 Probability
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 informationSTAT 430/510 Probability Lecture 12: Central Limit Theorem and Exponential Distribution
STAT 430/510 Probability Lecture 12: Central Limit Theorem and Exponential Distribution Pengyuan (Penelope) Wang June 15, 2011 Review Discussed Uniform Distribution and Normal Distribution Normal Approximation
More informationt x 1 e t dt, and simplify the answer when possible (for example, when r is a positive even number). In particular, confirm that EX 4 = 3.
Mathematical Statistics: Homewor problems General guideline. While woring outside the classroom, use any help you want, including people, computer algebra systems, Internet, and solution manuals, but mae
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2018 Examinations Subject CT3 Probability and Mathematical Statistics Core Technical Syllabus 1 June 2017 Aim The
More informationIntroduction to Probability and Statistics Slides 3 Chapter 3
Introduction to Probability and Statistics Slides 3 Chapter 3 Ammar M. Sarhan, asarhan@mathstat.dal.ca Department of Mathematics and Statistics, Dalhousie University Fall Semester 2008 Dr. Ammar M. Sarhan
More informationRandom Variables Example:
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More information15 Discrete Distributions
Lecture Note 6 Special Distributions (Discrete and Continuous) MIT 4.30 Spring 006 Herman Bennett 5 Discrete Distributions We have already seen the binomial distribution and the uniform distribution. 5.
More informationSTAT 302 Introduction to Probability Learning Outcomes. Textbook: A First Course in Probability by Sheldon Ross, 8 th ed.
STAT 302 Introduction to Probability Learning Outcomes Textbook: A First Course in Probability by Sheldon Ross, 8 th ed. Chapter 1: Combinatorial Analysis Demonstrate the ability to solve combinatorial
More informationKnown probability distributions
Known probability distributions Engineers frequently wor with data that can be modeled as one of several nown probability distributions. Being able to model the data allows us to: model real systems design
More informationDeccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY. SECOND YEAR B.Sc. SEMESTER - III
Deccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY SECOND YEAR B.Sc. SEMESTER - III SYLLABUS FOR S. Y. B. Sc. STATISTICS Academic Year 07-8 S.Y. B.Sc. (Statistics)
More information3 Continuous Random Variables
Jinguo Lian Math437 Notes January 15, 016 3 Continuous Random Variables Remember that discrete random variables can take only a countable number of possible values. On the other hand, a continuous random
More informationStatistics, Data Analysis, and Simulation SS 2013
Statistics, Data Analysis, and Simulation SS 213 8.128.73 Statistik, Datenanalyse und Simulation Dr. Michael O. Distler Mainz, 23. April 213 What we ve learned so far Fundamental
More informationAdvanced Herd Management Probabilities and distributions
Advanced Herd Management Probabilities and distributions Anders Ringgaard Kristensen Slide 1 Outline Probabilities Conditional probabilities Bayes theorem Distributions Discrete Continuous Distribution
More informationCOVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS
COVENANT UNIVERSITY NIGERIA TUTORIAL KIT OMEGA SEMESTER PROGRAMME: ECONOMICS COURSE: CBS 221 DISCLAIMER The contents of this document are intended for practice and leaning purposes at the undergraduate
More informationFIRST YEAR EXAM Monday May 10, 2010; 9:00 12:00am
FIRST YEAR EXAM Monday May 10, 2010; 9:00 12:00am NOTES: PLEASE READ CAREFULLY BEFORE BEGINNING EXAM! 1. Do not write solutions on the exam; please write your solutions on the paper provided. 2. Put the
More informationHypothesis testing Goodness of fit Multicollinearity Prediction. Applied Statistics. Lecturer: Serena Arima
Applied Statistics Lecturer: Serena Arima Hypothesis testing for the linear model Under the Gauss-Markov assumptions and the normality of the error terms, we saw that β N(β, σ 2 (X X ) 1 ) and hence s
More information4 Hypothesis testing. 4.1 Types of hypothesis and types of error 4 HYPOTHESIS TESTING 49
4 HYPOTHESIS TESTING 49 4 Hypothesis testing In sections 2 and 3 we considered the problem of estimating a single parameter of interest, θ. In this section we consider the related problem of testing whether
More informationContinuous random variables
Continuous random variables Continuous r.v. s take an uncountably infinite number of possible values. Examples: Heights of people Weights of apples Diameters of bolts Life lengths of light-bulbs We cannot
More informationSubject CS1 Actuarial Statistics 1 Core Principles
Institute of Actuaries of India Subject CS1 Actuarial Statistics 1 Core Principles For 2019 Examinations Aim The aim of the Actuarial Statistics 1 subject is to provide a grounding in mathematical and
More informationGuidelines for Solving Probability Problems
Guidelines for Solving Probability Problems CS 1538: Introduction to Simulation 1 Steps for Problem Solving Suggested steps for approaching a problem: 1. Identify the distribution What distribution does
More informationCHAPTER 6 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS. 6.2 Normal Distribution. 6.1 Continuous Uniform Distribution
CHAPTER 6 SOME CONTINUOUS PROBABILITY DISTRIBUTIONS Recall that a continuous random variable X is a random variable that takes all values in an interval or a set of intervals. The distribution of a continuous
More informationPlotting data is one method for selecting a probability distribution. The following
Advanced Analytical Models: Over 800 Models and 300 Applications from the Basel II Accord to Wall Street and Beyond By Johnathan Mun Copyright 008 by Johnathan Mun APPENDIX C Understanding and Choosing
More informationCommon probability distributionsi Math 217 Probability and Statistics Prof. D. Joyce, Fall 2014
Introduction. ommon probability distributionsi Math 7 Probability and Statistics Prof. D. Joyce, Fall 04 I summarize here some of the more common distributions used in probability and statistics. Some
More informationBINOMIAL DISTRIBUTION
BINOMIAL DISTRIBUTION The binomial distribution is a particular type of discrete pmf. It describes random variables which satisfy the following conditions: 1 You perform n identical experiments (called
More informationDiscrete probability distributions
Discrete probability s BSAD 30 Dave Novak Fall 08 Source: Anderson et al., 05 Quantitative Methods for Business th edition some slides are directly from J. Loucks 03 Cengage Learning Covered so far Chapter
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More informationDr. Maddah ENMG 617 EM Statistics 10/15/12. Nonparametric Statistics (2) (Goodness of fit tests)
Dr. Maddah ENMG 617 EM Statistics 10/15/12 Nonparametric Statistics (2) (Goodness of fit tests) Introduction Probability models used in decision making (Operations Research) and other fields require fitting
More informationNotes 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(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationBasics on Probability. Jingrui He 09/11/2007
Basics on Probability Jingrui He 09/11/2007 Coin Flips You flip a coin Head with probability 0.5 You flip 100 coins How many heads would you expect Coin Flips cont. You flip a coin Head with probability
More informationChapter 3 Multiple Regression Complete Example
Department of Quantitative Methods & Information Systems ECON 504 Chapter 3 Multiple Regression Complete Example Spring 2013 Dr. Mohammad Zainal Review Goals After completing this lecture, you should be
More informationStat 5101 Notes: Brand Name Distributions
Stat 5101 Notes: Brand Name Distributions Charles J. Geyer February 14, 2003 1 Discrete Uniform Distribution DiscreteUniform(n). Discrete. Rationale Equally likely outcomes. The interval 1, 2,..., n of
More informationSTAT 135 Lab 5 Bootstrapping and Hypothesis Testing
STAT 135 Lab 5 Bootstrapping and Hypothesis Testing Rebecca Barter March 2, 2015 The Bootstrap Bootstrap Suppose that we are interested in estimating a parameter θ from some population with members x 1,...,
More informationCS 361: Probability & Statistics
February 26, 2018 CS 361: Probability & Statistics Random variables The discrete uniform distribution If every value of a discrete random variable has the same probability, then its distribution is called
More informationPractice Problems Section Problems
Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,
More informationDiscrete Probability Distribution
Shapes of binomial distributions Discrete Probability Distribution Week 11 For this activity you will use a web applet. Go to http://socr.stat.ucla.edu/htmls/socr_eperiments.html and choose Binomial coin
More informationSummary of Chapters 7-9
Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two
More informationChapter 12 - Lecture 2 Inferences about regression coefficient
Chapter 12 - Lecture 2 Inferences about regression coefficient April 19th, 2010 Facts about slope Test Statistic Confidence interval Hypothesis testing Test using ANOVA Table Facts about slope In previous
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 7 Inferences Based on Two Samples: Confidence Intervals & Tests of Hypotheses Content 1. Identifying the Target Parameter 2. Comparing Two Population Means:
More informationFirst Year Examination Department of Statistics, University of Florida
First Year Examination Department of Statistics, University of Florida August 20, 2009, 8:00 am - 2:00 noon Instructions:. You have four hours to answer questions in this examination. 2. You must show
More informationContinuous Random Variables and Continuous Distributions
Continuous Random Variables and Continuous Distributions Continuous Random Variables and Continuous Distributions Expectation & Variance of Continuous Random Variables ( 5.2) The Uniform Random Variable
More informationStat 5101 Notes: Brand Name Distributions
Stat 5101 Notes: Brand Name Distributions Charles J. Geyer September 5, 2012 Contents 1 Discrete Uniform Distribution 2 2 General Discrete Uniform Distribution 2 3 Uniform Distribution 3 4 General Uniform
More informationStatistics for scientists and engineers
Statistics for scientists and engineers February 0, 006 Contents Introduction. Motivation - why study statistics?................................... Examples..................................................3
More informationMA6451 PROBABILITY AND RANDOM PROCESSES
MA6451 PROBABILITY AND RANDOM PROCESSES UNIT I RANDOM VARIABLES 1.1 Discrete and continuous random variables 1. Show that the function is a probability density function of a random variable X. (Apr/May
More informationMathematical statistics
October 1 st, 2018 Lecture 11: Sufficient statistic Where are we? Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationWeek 1 Quantitative Analysis of Financial Markets Distributions A
Week 1 Quantitative Analysis of Financial Markets Distributions A Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
More information16.400/453J Human Factors Engineering. Design of Experiments II
J Human Factors Engineering Design of Experiments II Review Experiment Design and Descriptive Statistics Research question, independent and dependent variables, histograms, box plots, etc. Inferential
More informationChapter 3. Discrete Random Variables and Their Probability Distributions
Chapter 3. Discrete Random Variables and Their Probability Distributions 1 3.4-3 The Binomial random variable The Binomial random variable is related to binomial experiments (Def 3.6) 1. The experiment
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests
Statistics for Managers Using Microsoft Excel/SPSS Chapter 8 Fundamentals of Hypothesis Testing: One-Sample Tests 1999 Prentice-Hall, Inc. Chap. 8-1 Chapter Topics Hypothesis Testing Methodology Z Test
More informationExponential & Gamma Distributions
Exponential & Gamma Distributions Engineering Statistics Section 4.4 Josh Engwer TTU 7 March 26 Josh Engwer (TTU) Exponential & Gamma Distributions 7 March 26 / 2 PART I PART I: EXPONENTIAL DISTRIBUTION
More informationerrors every 1 hour unless he falls asleep, in which case he just reports the total errors
I. First Definition of a Poisson Process A. Definition: Poisson process A Poisson Process {X(t), t 0} with intensity λ > 0 is a counting process with the following properties. INDEPENDENT INCREMENTS. For
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationRelationship between probability set function and random variable - 2 -
2.0 Random Variables A rat is selected at random from a cage and its sex is determined. The set of possible outcomes is female and male. Thus outcome space is S = {female, male} = {F, M}. If we let X be
More informationHypothesis Testing. ECE 3530 Spring Antonio Paiva
Hypothesis Testing ECE 3530 Spring 2010 Antonio Paiva What is hypothesis testing? A statistical hypothesis is an assertion or conjecture concerning one or more populations. To prove that a hypothesis is
More informationMaster s Written Examination - Solution
Master s Written Examination - Solution Spring 204 Problem Stat 40 Suppose X and X 2 have the joint pdf f X,X 2 (x, x 2 ) = 2e (x +x 2 ), 0 < x < x 2
More informationMathematical Statistics 1 Math A 6330
Mathematical Statistics 1 Math A 6330 Chapter 3 Common Families of Distributions Mohamed I. Riffi Department of Mathematics Islamic University of Gaza September 28, 2015 Outline 1 Subjects of Lecture 04
More informationS n = x + X 1 + X X n.
0 Lecture 0 0. Gambler Ruin Problem Let X be a payoff if a coin toss game such that P(X = ) = P(X = ) = /2. Suppose you start with x dollars and play the game n times. Let X,X 2,...,X n be payoffs in each
More informationDefinition: A random variable X is a real valued function that maps a sample space S into the space of real numbers R. X : S R
Random Variables Definition: A random variable X is a real valued function that maps a sample space S into the space of real numbers R. X : S R As such, a random variable summarizes the outcome of an experiment
More informationChapter 9 Inferences from Two Samples
Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review
More informationProbability: Why do we care? Lecture 2: Probability and Distributions. Classical Definition. What is Probability?
Probability: Why do we care? Lecture 2: Probability and Distributions Sandy Eckel seckel@jhsph.edu 22 April 2008 Probability helps us by: Allowing us to translate scientific questions into mathematical
More informationLecture 13. Poisson Distribution. Text: A Course in Probability by Weiss 5.5. STAT 225 Introduction to Probability Models February 16, 2014
Lecture 13 Text: A Course in Probability by Weiss 5.5 STAT 225 Introduction to Probability Models February 16, 2014 Whitney Huang Purdue University 13.1 Agenda 1 2 3 13.2 Review So far, we have seen discrete
More information(Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3)
3 Probability Distributions (Ch 3.4.1, 3.4.2, 4.1, 4.2, 4.3) Probability Distribution Functions Probability distribution function (pdf): Function for mapping random variables to real numbers. Discrete
More informationSolutions to First Midterm Exam, Stat 371, Spring those values: = Or, we can use Rule 6: = 0.63.
Solutions to First Midterm Exam, Stat 371, Spring 2010 There are two, three or four versions of each question. The questions on your exam comprise a mix of versions. As a result, when you examine the solutions
More informationRMSC 2001 Introduction to Risk Management
RMSC 2001 Introduction to Risk Management Tutorial 4 (2011/12) 1 February 20, 2012 Outline: 1. Failure Time 2. Loss Frequency 3. Loss Severity 4. Aggregate Claim ====================================================
More informationHYPOTHESIS TESTING: FREQUENTIST APPROACH.
HYPOTHESIS TESTING: FREQUENTIST APPROACH. These notes summarize the lectures on (the frequentist approach to) hypothesis testing. You should be familiar with the standard hypothesis testing from previous
More informationSTAT 3610: Review of Probability Distributions
STAT 3610: Review of Probability Distributions Mark Carpenter Professor of Statistics Department of Mathematics and Statistics August 25, 2015 Support of a Random Variable Definition The support of a random
More informationDiscrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan Introduction The markets can be thought of as a complex interaction of a large number of random processes,
More informationProbability and Distributions
Probability and Distributions What is a statistical model? A statistical model is a set of assumptions by which the hypothetical population distribution of data is inferred. It is typically postulated
More informationRandom variables. DS GA 1002 Probability and Statistics for Data Science.
Random variables DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Motivation Random variables model numerical quantities
More informationSTAT:5100 (22S:193) Statistical Inference I
STAT:5100 (22S:193) Statistical Inference I Week 10 Luke Tierney University of Iowa Fall 2015 Luke Tierney (U Iowa) STAT:5100 (22S:193) Statistical Inference I Fall 2015 1 Monday, October 26, 2015 Recap
More informationECE 313 Probability with Engineering Applications Fall 2000
Exponential random variables Exponential random variables arise in studies of waiting times, service times, etc X is called an exponential random variable with parameter λ if its pdf is given by f(u) =
More informationStat 135, Fall 2006 A. Adhikari HOMEWORK 6 SOLUTIONS
Stat 135, Fall 2006 A. Adhikari HOMEWORK 6 SOLUTIONS 1a. Under the null hypothesis X has the binomial (100,.5) distribution with E(X) = 50 and SE(X) = 5. So P ( X 50 > 10) is (approximately) two tails
More informationThree hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER.
Three hours To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER EXTREME VALUES AND FINANCIAL RISK Examiner: Answer QUESTION 1, QUESTION
More informationChapter 5. Statistical Models in Simulations 5.1. Prof. Dr. Mesut Güneş Ch. 5 Statistical Models in Simulations
Chapter 5 Statistical Models in Simulations 5.1 Contents Basic Probability Theory Concepts Discrete Distributions Continuous Distributions Poisson Process Empirical Distributions Useful Statistical Models
More information, 0 x < 2. a. Find the probability that the text is checked out for more than half an hour but less than an hour. = (1/2)2
Math 205 Spring 206 Dr. Lily Yen Midterm 2 Show all your work Name: 8 Problem : The library at Capilano University has a copy of Math 205 text on two-hour reserve. Let X denote the amount of time the text
More informationExam 2 Practice Questions, 18.05, Spring 2014
Exam 2 Practice Questions, 18.05, Spring 2014 Note: This is a set of practice problems for exam 2. The actual exam will be much shorter. Within each section we ve arranged the problems roughly in order
More informationAn-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)
An-Najah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 3 Discrete Random
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Chapter 9 Hypothesis Testing: Single Population Ch. 9-1 9.1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population
More informationCHAPTER 6. 1, if n =1, 2p(1 p), if n =2, n (1 p) n 1 n p + p n 1 (1 p), if n =3, 4, 5,... var(d) = 4var(R) =4np(1 p).
CHAPTER 6 Solution to Problem 6 (a) The random variable R is binomial with parameters p and n Hence, ( ) n p R(r) = ( p) n r p r, for r =0,,,,n, r E[R] = np, and var(r) = np( p) (b) Let A be the event
More informationARCONES MANUAL FOR THE SOA EXAM P/CAS EXAM 1, PROBABILITY, SPRING 2010 EDITION.
A self published manuscript ARCONES MANUAL FOR THE SOA EXAM P/CAS EXAM 1, PROBABILITY, SPRING 21 EDITION. M I G U E L A R C O N E S Miguel A. Arcones, Ph. D. c 28. All rights reserved. Author Miguel A.
More informationMasters Comprehensive Examination Department of Statistics, University of Florida
Masters Comprehensive Examination Department of Statistics, University of Florida May 10, 2002, 8:00am - 12:00 noon Instructions: 1. You have four hours to answer questions in this examination. 2. There
More informationSalt Lake Community College MATH 1040 Final Exam Fall Semester 2011 Form E
Salt Lake Community College MATH 1040 Final Exam Fall Semester 011 Form E Name Instructor Time Limit: 10 minutes Any hand-held calculator may be used. Computers, cell phones, or other communication devices
More informationContinuous random variables
Continuous random variables Can take on an uncountably infinite number of values Any value within an interval over which the variable is definied has some probability of occuring This is different from
More informationLecture 2: Conjugate priors
(Spring ʼ) Lecture : Conjugate priors Julia Hockenmaier juliahmr@illinois.edu Siebel Center http://www.cs.uiuc.edu/class/sp/cs98jhm The binomial distribution If p is the probability of heads, the probability
More information