Will Murray s Probability, XXXII. Moment-Generating Functions 1. We want to study functions of them:
|
|
- Myles Charles
- 5 years ago
- Views:
Transcription
1 Will Murray s Probability, XXXII. Moment-Generating Functions XXXII. Moment-Generating Functions Premise We have several random variables, Y, Y, etc. We want to study functions of them: U (Y,..., Y n ). Before, we calculated the mean of U and the variance, but that s not enough to determine the whole distribution of U. Goal We want to find the full distribution function F U (u) := P (U u). Then we can find the density function f U (u) = F U(u). We can calculate probabilities: P (a U b) = b a f U (u) du = F U (b) F U (a) Three methods. Distribution functions. (Two lectures ago, using geometric methods from Calculus III.). Transformations. (Previous lecture, using methods from Calculus I.)
2 Will Murray s Probability, XXXII. Moment-Generating Functions 3. Moment-generating functions. (This lecture.) Review of Moment-Generating Functions Recall: The moment-generating function for a random variable Y is m Y (t) := E ( e ty ). The MGF is a function of t (not y). See previous lecture on MGFs. MGFs for the Discrete Distributions Distribution Binomial Geometric Negative binomial Hypergeometric MGF [ pe t + ( p) ] n pe t ( p)e t [ pe t ( p)e t ] r No closed-form MGF. Poisson e λ(et )
3 Will Murray s Probability, XXXII. Moment-Generating Functions 3 All are functions of t. In the first three, we could substitute q := p. MGFs for the Continuous Distributions Distribution Uniform MGF e tθ e tθ t (θ θ ) Normal e µt+ t σ Gamma Exponential ( βt) α ( βt) Chi-square ( t) ν Beta No closed-form MGF. Note that exponential is just gamma with α :=, and chi-square is gamma with α := ν and β :=. Useful Formulas with MGFs Let Z := ay + b. Then m Z (t) = e bt m Y (at).
4 Will Murray s Probability, XXXII. Moment-Generating Functions 4 Suppose Y and Y are independent variables and Z := Y + Y. Then m Z (t) = m Y (t)m Y (t) How to use MGFs Given a function U (Y,..., Y n ), find its MGF m U (t). Use the useful formulas on the previous slide. Then compare it against your charts to see if you recognize it as a known distribution. Example I Let Y be a standard normal variable. Find the density function of U := Y.
5 Will Murray s Probability, XXXII. Moment-Generating Functions 5 m Y (t) := E := Combine exponents: = = ty ] [e π π π e ty e y dy e y ( t) dy e y ( t) dy To simplify this, recall that σ e y σ dy = π (normal density) = for any temporary σ. Take temporary σ := = σ σ π = σ t : = ( t) e y σ dy From your chart of mgfs, this is chi-square with ν =, so we have the density: Gamma : f(y) := yα e y β β α Γ(α), 0 y < χ is gamma with α := ν, β :=. ( ) f U (u) = u e u ( Γ u > 0 Γ = ), π = u e u π, u > 0 That s one reason why chi-square is important.
6 Will Murray s Probability, XXXII. Moment-Generating Functions 6 Example II Let Y, Y be independent standard normal variables. Find the density function of U := Y + Y. m U (t) = m Y +Y (t) = m Y (t)m Y (t) = ( t) ( t) from above = ( t) Looking at the chart, this is chi-square with ν =. Gamma : f(y) := yα e y β β α Γ(α), 0 y < χ is gamma with α := ν, β :=. So f U (u) = e u, u > 0. (It s also exponential with β =.) n In general, if Z,..., Z n N(0, ), then χ (n). i= That s one reason why chi-square is important. Z i Example III Let Y,..., Y r be independent binomial variables representing n,..., n r flips of a coin that comes up heads with probability p. Find the probability
7 Will Murray s Probability, XXXII. Moment-Generating Functions 7 function of U := Y + + Y r. Note that 0 u n + + n r. m Yi (t) = [ pe t + ( p) ] n i m U (t) := m Y + +Y r (t) = m Y (t) m Yr (t) = [ pe t + ( p) ] n [pe t + ( p) ] n r = [ pe t + ( p) ] n + +n r This is binomial with probability p and n := n + + n r, so ( ) n + + n r p(u) = p u q n + +n r y, 0 u n + + n r. u Example IV Let Y and Y be independent Poisson variables with means λ and λ. Find the probability function of U := Y + Y.
8 Will Murray s Probability, XXXII. Moment-Generating Functions 8 m Yi (t) = e λ i(e t ) m U (t) := m Y +Y (t) = m Y (t)m Y (t) = e λ (e t ) e λ (e t ) = e (λ +λ )(e t ) This is Poisson with mean λ + λ. (If you expect to see λ cars at an intersection and λ trucks, you expect to see λ + λ vehicles total.) p(y) = λy e λ y! p(u) = (λ + λ ) u e λ λ, 0 u < u! Example V Let Y,..., Y n be independent normal variables, each with mean µ and variance σ. Find the distribution of Y := n (Y + + Y n ).
9 Will Murray s Probability, XXXII. Moment-Generating Functions 9 Let Y := Y + + Y n. m Y (t) = m Y (t) m Yn (t) = ) (e µt+ σ t n = e µnt+ σ t n m ay +b (t) = e bt m Y (at) ( ) t m Y = m Y = e µt+ σ t n n This is a normal distribution with mean µ, variance σ n. Example VI Let Y and Y be independent exponential variables, each with mean 3. Find the density function of U := Y + Y.
10 Will Murray s Probability, XXXII. Moment-Generating Functions 0 m Yi (t) = ( 3t) m U (t) := m Y +Y (t) = m Y (t)m Y (t) = ( 3t) ( 3t) = ( 3t) This is gamma with α =, β = 3: f U (u) = uα e u β β α Γ(α) = ue u 3 3 Γ() = 9 ue u 3, 0 u <
Contents 1. Contents
Contents 1 Contents 6 Distributions of Functions of Random Variables 2 6.1 Transformation of Discrete r.v.s............. 3 6.2 Method of Distribution Functions............. 6 6.3 Method of Transformations................
More informationReview for the previous lecture
Lecture 1 and 13 on BST 631: Statistical Theory I Kui Zhang, 09/8/006 Review for the previous lecture Definition: Several discrete distributions, including discrete uniform, hypergeometric, Bernoulli,
More informationLecture 1: August 28
36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 1: August 28 Our broad goal for the first few lectures is to try to understand the behaviour of sums of independent random
More informationBMIR Lecture Series on Probability and Statistics Fall 2015 Discrete RVs
Lecture #7 BMIR Lecture Series on Probability and Statistics Fall 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University 7.1 Function of Single Variable Theorem
More informationLecture 5: Moment generating functions
Lecture 5: Moment generating functions Definition 2.3.6. The moment generating function (mgf) of a random variable X is { x e tx f M X (t) = E(e tx X (x) if X has a pmf ) = etx f X (x)dx if X has a pdf
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 informationChapter 3 Common Families of Distributions
Lecture 9 on BST 631: Statistical Theory I Kui Zhang, 9/3/8 and 9/5/8 Review for the previous lecture Definition: Several commonly used discrete distributions, including discrete uniform, hypergeometric,
More information[Chapter 6. Functions of Random Variables]
[Chapter 6. Functions of Random Variables] 6.1 Introduction 6.2 Finding the probability distribution of a function of random variables 6.3 The method of distribution functions 6.5 The method of Moment-generating
More informationi=1 k i=1 g i (Y )] = k
Math 483 EXAM 2 covers 2.4, 2.5, 2.7, 2.8, 3.1, 3.2, 3.3, 3.4, 3.8, 3.9, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.9, 5.1, 5.2, and 5.3. The exam is on Thursday, Oct. 13. You are allowed THREE SHEETS OF NOTES and
More informationAPPM/MATH 4/5520 Solutions to Exam I Review Problems. f X 1,X 2. 2e x 1 x 2. = x 2
APPM/MATH 4/5520 Solutions to Exam I Review Problems. (a) f X (x ) f X,X 2 (x,x 2 )dx 2 x 2e x x 2 dx 2 2e 2x x was below x 2, but when marginalizing out x 2, we ran it over all values from 0 to and so
More informationLIST OF FORMULAS FOR STK1100 AND STK1110
LIST OF FORMULAS FOR STK1100 AND STK1110 (Version of 11. November 2015) 1. Probability Let A, B, A 1, A 2,..., B 1, B 2,... be events, that is, subsets of a sample space Ω. a) Axioms: A probability function
More information, find P(X = 2 or 3) et) 5. )px (1 p) n x x = 0, 1, 2,..., n. 0 elsewhere = 40
Assignment 4 Fall 07. Exercise 3.. on Page 46: If the mgf of a rom variable X is ( 3 + 3 et) 5, find P(X or 3). Since the M(t) of X is ( 3 + 3 et) 5, X has a binomial distribution with n 5, p 3. The probability
More informationSampling Distributions
Sampling Distributions In statistics, a random sample is a collection of independent and identically distributed (iid) random variables, and a sampling distribution is the distribution of a function of
More informationStat 515 Midterm Examination II April 4, 2016 (7:00 p.m. - 9:00 p.m.)
Name: Section: Stat 515 Midterm Examination II April 4, 2016 (7:00 p.m. - 9:00 p.m.) The total score is 120 points. Instructions: There are 10 questions. Please circle 8 problems below that you want to
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 informationAPPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679
APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared
More informationDistributions of Functions of Random Variables. 5.1 Functions of One Random Variable
Distributions of Functions of Random Variables 5.1 Functions of One Random Variable 5.2 Transformations of Two Random Variables 5.3 Several Random Variables 5.4 The Moment-Generating Function Technique
More informationContinuous Distributions
A normal distribution and other density functions involving exponential forms play the most important role in probability and statistics. They are related in a certain way, as summarized in a diagram later
More information1 Solution to Problem 2.1
Solution to Problem 2. I incorrectly worked this exercise instead of 2.2, so I decided to include the solution anyway. a) We have X Y /3, which is a - function. It maps the interval, ) where X lives) onto
More information7 Random samples and sampling distributions
7 Random samples and sampling distributions 7.1 Introduction - random samples We will use the term experiment in a very general way to refer to some process, procedure or natural phenomena that produces
More informationLecture The Sample Mean and the Sample Variance Under Assumption of Normality
Math 408 - Mathematical Statistics Lecture 13-14. The Sample Mean and the Sample Variance Under Assumption of Normality February 20, 2013 Konstantin Zuev (USC) Math 408, Lecture 13-14 February 20, 2013
More informationSOLUTION FOR HOMEWORK 12, STAT 4351
SOLUTION FOR HOMEWORK 2, STAT 435 Welcome to your 2th homework. It looks like this is the last one! As usual, try to find mistakes and get extra points! Now let us look at your problems.. Problem 7.22.
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 informationPage Max. Possible Points Total 100
Math 3215 Exam 2 Summer 2014 Instructor: Sal Barone Name: GT username: 1. No books or notes are allowed. 2. You may use ONLY NON-GRAPHING and NON-PROGRAMABLE scientific calculators. All other electronic
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 information1, 0 r 1 f R (r) = 0, otherwise. 1 E(R 2 ) = r 2 f R (r)dr = r 2 3
STAT 5 4.43. We are given that a circle s radius R U(, ). the pdf of R is {, r f R (r), otherwise. The area of the circle is A πr. The mean of A is E(A) E(πR ) πe(r ). The second moment of R is ( ) r E(R
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2017 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Exercise 4.1 Let X be a random variable with p(x)
More informationSuppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8.
Suppose that you have three coins. Coin A is fair, coin B shows heads with probability 0.6 and coin C shows heads with probability 0.8. Coin A is flipped until a head appears, then coin B is flipped until
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 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 informationStat 5101 Lecture Slides: Deck 8 Dirichlet Distribution. Charles J. Geyer School of Statistics University of Minnesota
Stat 5101 Lecture Slides: Deck 8 Dirichlet Distribution Charles J. Geyer School of Statistics University of Minnesota 1 The Dirichlet Distribution The Dirichlet Distribution is to the beta distribution
More information18.440: Lecture 28 Lectures Review
18.440: Lecture 28 Lectures 18-27 Review Scott Sheffield MIT Outline Outline It s the coins, stupid Much of what we have done in this course can be motivated by the i.i.d. sequence X i where each X i is
More informationSampling Distributions
In statistics, a random sample is a collection of independent and identically distributed (iid) random variables, and a sampling distribution is the distribution of a function of random sample. For example,
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationIntroduction to Probability
Introduction to Probability Math 353, Section 1 Fall 212 Homework 9 Solutions 1. During a season, a basketball team plays 7 home games and 6 away games. The coach estimates at the beginning of the season
More informationIEOR 3106: Introduction to Operations Research: Stochastic Models. Professor Whitt. SOLUTIONS to Homework Assignment 2
IEOR 316: Introduction to Operations Research: Stochastic Models Professor Whitt SOLUTIONS to Homework Assignment 2 More Probability Review: In the Ross textbook, Introduction to Probability Models, read
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 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 informationLimiting Distributions
Limiting Distributions We introduce the mode of convergence for a sequence of random variables, and discuss the convergence in probability and in distribution. The concept of convergence leads us to the
More informationLimiting Distributions
We introduce the mode of convergence for a sequence of random variables, and discuss the convergence in probability and in distribution. The concept of convergence leads us to the two fundamental results
More informationLecture 17: The Exponential and Some Related Distributions
Lecture 7: The Exponential and Some Related Distributions. Definition Definition: A continuous random variable X is said to have the exponential distribution with parameter if the density of X is e x if
More informationStat 366 A1 (Fall 2006) Midterm Solutions (October 23) page 1
Stat 366 A1 Fall 6) Midterm Solutions October 3) page 1 1. The opening prices per share Y 1 and Y measured in dollars) of two similar stocks are independent random variables, each with a density function
More informationThe Multivariate Normal Distribution 1
The Multivariate Normal Distribution 1 STA 302 Fall 2017 1 See last slide for copyright information. 1 / 40 Overview 1 Moment-generating Functions 2 Definition 3 Properties 4 χ 2 and t distributions 2
More informationChapter 8. Some Approximations to Probability Distributions: Limit Theorems
Chapter 8. Some Approximations to Probability Distributions: Limit Theorems Sections 8.2 -- 8.3: Convergence in Probability and in Distribution Jiaping Wang Department of Mathematical Science 04/22/2013,
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 informationTest Problems for Probability Theory ,
1 Test Problems for Probability Theory 01-06-16, 010-1-14 1. Write down the following probability density functions and compute their moment generating functions. (a) Binomial distribution with mean 30
More informationA Few Special Distributions and Their Properties
A Few Special Distributions and Their Properties Econ 690 Purdue University Justin L. Tobias (Purdue) Distributional Catalog 1 / 20 Special Distributions and Their Associated Properties 1 Uniform Distribution
More informationChapter 5. Means and Variances
1 Chapter 5 Means and Variances Our discussion of probability has taken us from a simple classical view of counting successes relative to total outcomes and has brought us to the idea of a probability
More informationSTA2603/205/1/2014 /2014. ry II. Tutorial letter 205/1/
STA263/25//24 Tutorial letter 25// /24 Distribution Theor ry II STA263 Semester Department of Statistics CONTENTS: Examination preparation tutorial letterr Solutions to Assignment 6 2 Dear Student, This
More informationSome Special Distributions (Hogg Chapter Three)
Some Special Distributions Hogg Chapter Three STAT 45-: Mathematical Statistics I Fall Semester 5 Contents Binomial & Related Distributions. The Binomial Distribution............... Some advice on calculating
More informationBrief 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 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 informationStatistics 1B. Statistics 1B 1 (1 1)
0. Statistics 1B Statistics 1B 1 (1 1) 0. Lecture 1. Introduction and probability review Lecture 1. Introduction and probability review 2 (1 1) 1. Introduction and probability review 1.1. What is Statistics?
More information3. Probability and Statistics
FE661 - Statistical Methods for Financial Engineering 3. Probability and Statistics Jitkomut Songsiri definitions, probability measures conditional expectations correlation and covariance some important
More informationPROBABILITY DISTRIBUTIONS. J. Elder CSE 6390/PSYC 6225 Computational Modeling of Visual Perception
PROBABILITY DISTRIBUTIONS Credits 2 These slides were sourced and/or modified from: Christopher Bishop, Microsoft UK Parametric Distributions 3 Basic building blocks: Need to determine given Representation:
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 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 information2) Are the events A and B independent? Say why or why not [Sol] No : P (A B) = 0.12 is not equal to P (A) P (B) = =
Stat 516 (Spring 2010) Hw 1 (due Feb. 2, Tuesday) Question 1 Suppose P (A) =0.45, P (B) = 0.32 and P (Ā B) = 0.20. 1) Find P (A B) [Sol] Since P (B) =P (A B)+P (Ā B), P (A B) =P (B) P (Ā B) =0.32 0.20
More information6. Bernoulli Trials and the Poisson Process
1 of 5 7/16/2009 7:09 AM Virtual Laboratories > 14. The Poisson Process > 1 2 3 4 5 6 7 6. Bernoulli Trials and the Poisson Process Basic Comparison In some sense, the Poisson process is a continuous time
More informationSome Special Distributions (Hogg Chapter Three)
Some Special Distributions Hogg Chapter Three STAT 45-: Mathematical Statistics I Fall Semester 3 Contents Binomial & Related Distributions The Binomial Distribution Some advice on calculating n 3 k Properties
More informationChapter 4 Multiple Random Variables
Review for the previous lecture Theorems and Examples: How to obtain the pmf (pdf) of U = g ( X Y 1 ) and V = g ( X Y) Chapter 4 Multiple Random Variables Chapter 43 Bivariate Transformations Continuous
More information18.440: Lecture 28 Lectures Review
18.440: Lecture 28 Lectures 17-27 Review Scott Sheffield MIT 1 Outline Continuous random variables Problems motivated by coin tossing Random variable properties 2 Outline Continuous random variables Problems
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 informationMore Spectral Clustering and an Introduction to Conjugacy
CS8B/Stat4B: Advanced Topics in Learning & Decision Making More Spectral Clustering and an Introduction to Conjugacy Lecturer: Michael I. Jordan Scribe: Marco Barreno Monday, April 5, 004. Back to spectral
More informationContinuous Distributions
Chapter 3 Continuous Distributions 3.1 Continuous-Type Data In Chapter 2, we discuss random variables whose space S contains a countable number of outcomes (i.e. of discrete type). In Chapter 3, we study
More informationTHE QUEEN S UNIVERSITY OF BELFAST
THE QUEEN S UNIVERSITY OF BELFAST 0SOR20 Level 2 Examination Statistics and Operational Research 20 Probability and Distribution Theory Wednesday 4 August 2002 2.30 pm 5.30 pm Examiners { Professor R M
More informationJoint Probability Distributions, Correlations
Joint Probability Distributions, Correlations What we learned so far Events: Working with events as sets: union, intersection, etc. Some events are simple: Head vs Tails, Cancer vs Healthy Some are more
More informationStatistics 3657 : Moment Generating Functions
Statistics 3657 : Moment Generating Functions A useful tool for studying sums of independent random variables is generating functions. course we consider moment generating functions. In this Definition
More informationLecture 3. Univariate Bayesian inference: conjugate analysis
Summary Lecture 3. Univariate Bayesian inference: conjugate analysis 1. Posterior predictive distributions 2. Conjugate analysis for proportions 3. Posterior predictions for proportions 4. Conjugate analysis
More informationMathematics 375 Probability and Statistics I Final Examination Solutions December 14, 2009
Mathematics 375 Probability and Statistics I Final Examination Solutions December 4, 9 Directions Do all work in the blue exam booklet. There are possible regular points and possible Extra Credit points.
More informationExpectation, variance and moments
Expectation, variance and moments John Appleby Contents Expectation and variance Examples 3 Moments and the moment generating function 4 4 Examples of moment generating functions 5 5 Concluding remarks
More informationOrder Statistics. The order statistics of a set of random variables X 1, X 2,, X n are the same random variables arranged in increasing order.
Order Statistics The order statistics of a set of random variables 1, 2,, n are the same random variables arranged in increasing order. Denote by (1) = smallest of 1, 2,, n (2) = 2 nd smallest of 1, 2,,
More information3 Conditional Expectation
3 Conditional Expectation 3.1 The Discrete case Recall that for any two events E and F, the conditional probability of E given F is defined, whenever P (F ) > 0, by P (E F ) P (E)P (F ). P (F ) Example.
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4.1 (*. Let independent variables X 1,..., X n have U(0, 1 distribution. Show that for every x (0, 1, we have P ( X (1 < x 1 and P ( X (n > x 1 as n. Ex. 4.2 (**. By using induction
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 16 June 24th, 2009 Review Sum of Independent Normal Random Variables Sum of Independent Poisson Random Variables Sum of Independent Binomial Random Variables Conditional
More informationSTA 2101/442 Assignment 3 1
STA 2101/442 Assignment 3 1 These questions are practice for the midterm and final exam, and are not to be handed in. 1. Suppose X 1,..., X n are a random sample from a distribution with mean µ and variance
More informationPartial Solutions for h4/2014s: Sampling Distributions
27 Partial Solutions for h4/24s: Sampling Distributions ( Let X and X 2 be two independent random variables, each with the same probability distribution given as follows. f(x 2 e x/2, x (a Compute the
More informationChapter 4. Chapter 4 sections
Chapter 4 sections 4.1 Expectation 4.2 Properties of Expectations 4.3 Variance 4.4 Moments 4.5 The Mean and the Median 4.6 Covariance and Correlation 4.7 Conditional Expectation SKIP: 4.8 Utility Expectation
More informationOne Parameter Models
One Parameter Models p. 1/2 One Parameter Models September 22, 2010 Reading: Hoff Chapter 3 One Parameter Models p. 2/2 Highest Posterior Density Regions Find Θ 1 α = {θ : p(θ Y ) h α } such that P (θ
More informationProbability Models. 4. What is the definition of the expectation of a discrete random variable?
1 Probability Models The list of questions below is provided in order to help you to prepare for the test and exam. It reflects only the theoretical part of the course. You should expect the questions
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 informationDiscrete Distributions
A simplest example of random experiment is a coin-tossing, formally called Bernoulli trial. It happens to be the case that many useful distributions are built upon this simplest form of experiment, whose
More informationIE 303 Discrete-Event Simulation
IE 303 Discrete-Event Simulation 1 L E C T U R E 5 : P R O B A B I L I T Y R E V I E W Review of the Last Lecture Random Variables Probability Density (Mass) Functions Cumulative Density Function Discrete
More informationProbability. Machine Learning and Pattern Recognition. Chris Williams. School of Informatics, University of Edinburgh. August 2014
Probability Machine Learning and Pattern Recognition Chris Williams School of Informatics, University of Edinburgh August 2014 (All of the slides in this course have been adapted from previous versions
More informationProbability Density Functions and the Normal Distribution. Quantitative Understanding in Biology, 1.2
Probability Density Functions and the Normal Distribution Quantitative Understanding in Biology, 1.2 1. Discrete Probability Distributions 1.1. The Binomial Distribution Question: You ve decided to flip
More informationTABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1
TABLE OF CONTENTS CHAPTER 1 COMBINATORIAL PROBABILITY 1 1.1 The Probability Model...1 1.2 Finite Discrete Models with Equally Likely Outcomes...5 1.2.1 Tree Diagrams...6 1.2.2 The Multiplication Principle...8
More informationBivariate Normal Distribution
.0. TWO-DIMENSIONAL RANDOM VARIABLES 47.0.7 Bivariate Normal Distribution Figure.: Bivariate Normal pdf Here we use matrix notation. A bivariate rv is treated as a random vector X X =. The expectation
More informationChapter 3 Single Random Variables and Probability Distributions (Part 1)
Chapter 3 Single Random Variables and Probability Distributions (Part 1) Contents What is a Random Variable? Probability Distribution Functions Cumulative Distribution Function Probability Density Function
More informationChapter 7: Special Distributions
This chater first resents some imortant distributions, and then develos the largesamle distribution theory which is crucial in estimation and statistical inference Discrete distributions The Bernoulli
More informationReview 1: STAT Mark Carpenter, Ph.D. Professor of Statistics Department of Mathematics and Statistics. August 25, 2015
Review : STAT 36 Mark Carpenter, Ph.D. Professor of Statistics Department of Mathematics and Statistics August 25, 25 Support of a Random Variable The support of a random variable, which is usually denoted
More informationPreface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of
Preface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of Probability Sampling Procedures Collection of Data Measures
More informationSeverity Models - Special Families of Distributions
Severity Models - Special Families of Distributions Sections 5.3-5.4 Stat 477 - Loss Models Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 1 / 1 Introduction Introduction Given that
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 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 informationELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random
More informationSolution-Midterm Examination - I STAT 421, Spring 2012
Solution-Midterm Examination - I STAT 4, Spring 0 Prof. Prem K. Goel. [0 points] Let X, X,..., X n be a random sample of size n from a Gamma population with β = and α = ν/, where ν is the unknown parameter.
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 informationTHE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA
THE ROYAL STATISTICAL SOCIETY 4 EXAINATIONS SOLUTIONS GRADUATE DIPLOA PAPER I STATISTICAL THEORY & ETHODS The Societ provides these solutions to assist candidates preparing for the examinations in future
More informationPractice Midterm 2 Partial Solutions
8.440 Practice Midterm Partial Solutions. (0 points) Let and Y be independent Poisson random variables with parameter. Compute the following. (Give a correct formula involving sums does not need to be
More informationM2S2 Statistical Modelling
UNIVERSITY OF LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2006 This paper is also taken for the relevant examination for the Associateship. M2S2 Statistical Modelling Date: Wednesday, 17th
More information