Point Estimation. Copyright Cengage Learning. All rights reserved.
|
|
- Chad Simon
- 6 years ago
- Views:
Transcription
1 6 Point Estimation Copyright Cengage Learning. All rights reserved.
2 6.2 Methods of Point Estimation Copyright Cengage Learning. All rights reserved.
3 Methods of Point Estimation The definition of unbiasedness does not in general indicate how unbiased estimators can be derived. We now discuss two constructive methods for obtaining point estimators: the method of moments and the method of maximum likelihood. By constructive we mean that the general definition of each type of estimator suggests explicitly how to obtain the estimator in any specific problem. 3
4 Methods of Point Estimation Although maximum likelihood estimators are generally preferable to moment estimators because of certain efficiency properties, they often require significantly more computation than do moment estimators. It is sometimes the case that these methods yield unbiased estimators. 4
5 The Method of Moments 5
6 The Method of Moments The basic idea of this method is to equate certain sample characteristics, such as the mean, to the corresponding population expected values. Then solving these equations for unknown parameter values yields the estimators. 6
7 The Method of Moments Definition Let X 1,..., X n be a random sample from a pmf or pdf f(x). For k = 1, 2, 3,..., the kth population moment, or kth moment of the distribution f(x), is E(X k ). The kth sample moment is Thus the first population moment is E(X) = µ, and the first sample moment is ΣX i /n = The second population and sample moments are E(X 2 ) and ΣX i2 /n, respectively. The population moments will be functions of any unknown parameters θ 1, θ 2,.... 7
8 The Method of Moments Definition Let X 1, X 2,..., X n be a random sample from a distribution with pmf or pdf f (x; θ 1,..., θ m ), where θ 1,..., θ m are parameters whose values are unknown. Then the moment estimators θ 1,..., θ m are obtained by equating the first m sample moments to the corresponding first m population moments and solving for θ 1,..., θ m. 8
9 The Method of Moments If, for example, m = 2, E(X) and E(X 2 ) will be functions of θ 1 and θ 2. Setting E(X) = (1/n) Σ X i (= ) and E(X 2 ) = (1/n) Σ X i 2 gives two equations in θ 1 and θ 2. The solution then defines the estimators. 9
10 Example 12 Let X 1, X 2,..., X n represent a random sample of service times of n customers at a certain facility, where the underlying distribution is assumed exponential with parameter λ. Since there is only one parameter to be estimated, the estimator is obtained by equating E(X) to. Since E(X) = 1/λ for an exponential distribution, this gives 1/λ = or λ = 1/. The moment estimator of λ is then λ 10
11 Maximum Likelihood Estimation 11
12 Maximum Likelihood Estimation The method of maximum likelihood was first introduced by R. A. Fisher, a geneticist and statistician, in the 1920s. Most statisticians recommend this method, at least when the sample size is large, since the resulting estimators have certain desirable efficiency properties. 12
13 Example 15 A sample of ten new bike helmets manufactured by a certain company is obtained. Upon testing, it is found that the first, third, and tenth helmets are flawed, whereas the others are not. Let p = P(flawed helmet), i.e., p is the proportion of all such helmets that are flawed. Define (Bernoulli) random variables X 1, X 2,..., X 10 by 13
14 Example 15 cont d Then for the obtained sample, X 1 = X 3 = X 10 = 1 and the other seven X i s are all zero. The probability mass function of any particular X i is, which becomes p if x i = 1 and 1 p when x i = 0. Now suppose that the conditions of various helmets are independent of one another. This implies that the X i s are independent, so their joint probability mass function is the product of the individual pmf s. 14
15 Example 15 cont d Thus the joint pmf evaluated at the observed X i s is f (x 1,..., x 10 ; p) = p(1 p)p... p = p 3 (1 p) 7 (6.4) Suppose that p =.25. Then the probability of observing the sample that we actually obtained is (.25) 3 (.75) 7 = If instead p =.50, then this probability is (.50) 3 (.50) 7 = For what value of p is the obtained sample most likely to have occurred? That is, for what value of p is the joint pmf (6.4) as large as it can be? What value of p maximizes (6.4)? 15
16 Example 15 cont d Figure 6.5(a) shows a graph of the likelihood (6.4) as a function of p. It appears that the graph reaches its peak above p =.3 = the proportion of flawed helmets in the sample. Graph of the likelihood (joint pmf) (6.4) from Example 15 Figure 6.5(a) 16
17 Example 15 cont d Figure 6.5(b) shows a graph of the natural logarithm of (6.4); since ln[g(u)] is a strictly increasing function of g(u), finding u to maximize the function g(u) is the same as finding u to maximize ln[g(u)]. Graph of the natural logarithm of the likelihood Figure 6.5(b) 17
18 Example 15 cont d We can verify our visual impression by using calculus to find the value of p that maximizes (6.4). Working with the natural log of the joint pmf is often easier than working with the joint pmf itself, since the joint pmf is typically a product so its logarithm will be a sum. Here ln[ f (x 1,..., x 10 ; p)] = ln[p 3 (1 p) 7 ] = 3ln(p) + 7ln(1 p) (6.5) 18
19 Example 15 cont d Thus [the (1) comes from the chain rule in calculus]. 19
20 Example 15 cont d Equating this derivative to 0 and solving for p gives 3(1 p) = 7p, from which 3 = 10p and so p = 3/10 =.30 as conjectured. That is, our point estimate is =.30. It is called the maximum likelihood estimate because it is the parameter value that maximizes the likelihood (joint pmf) of the observed sample. In general, the second derivative should be examined to make sure a maximum has been obtained, but here this is obvious from Figure
21 Example 15 cont d Suppose that rather than being told the condition of every helmet, we had only been informed that three of the ten were flawed. Then we would have the observed value of a binomial random variable X = the number of flawed helmets. The pmf of X is For x = 3, this becomes The binomial coefficient is irrelevant to the maximization, so again =
22 Maximum Likelihood Estimation The likelihood function tells us how likely the observed sample is as a function of the possible parameter values. Maximizing the likelihood gives the parameter values for which the observed sample is most likely to have been generated that is, the parameter values that agree most closely with the observed data. 22
23 Example 16 Suppose X 1, X 2,..., X n is a random sample from an exponential distribution with parameter λ. Because of independence, the likelihood function is a product of the individual pdf s: The natural logarithm of the likelihood function is ln[ f (x 1,..., x n ; λ)] = n ln(λ) λσx i 23
24 Example 16 cont d Equating (d/dλ)[ln(likelihood)] to zero results in n/λ Σx i = 0, or λ = n/σx i = Thus the mle is it is identical to the method of moments estimator [but it is not an unbiased estimator, since 24
25 Example 17 Let X 1,..., X n be a random sample from a normal distribution. The likelihood function is so 25
26 Example 17 cont d To find the maximizing values of µ and σ 2, we must take the partial derivatives of ln(f ) with respect to µ and σ 2, equate them to zero, and solve the resulting two equations. Omitting the details, the resulting mle s are The mle of σ 2 is not the unbiased estimator, so two different principles of estimation (unbiasedness and maximum likelihood) yield two different estimators. 26
27 Estimating Functions of Parameters 27
28 Estimating Functions of Parameters In Example 17, we obtained the mle of σ 2 when the underlying distribution is normal. The mle of σ =, as well as that of many other mle s, can be easily derived using the following proposition. Proposition The Invariance Principle Let θ 1, θ 2,..., θ m. be the mle s of the parameters Then the mle of any function h(θ 1, θ 2,..., θ m ) of these parameters is the function h( ) of the mle s. 28
29 Example 20 Example 17 continued In the normal case, the mle s of µ and σ 2 are To obtain the mle of the function substitute the mle s into the function: The mle of σ is not the sample standard deviation S, though they are close unless n is quite small. 29
30 Large Sample Behavior of the MLE 30
31 Large Sample Behavior of the MLE Although the principle of maximum likelihood estimation has considerable intuitive appeal, the following proposition provides additional rationale for the use of mle s. Proposition Under very general conditions on the joint distribution of the sample, when the sample size n is large, the maximum likelihood estimator of any parameter θ is approximately unbiased and has variance that is either as small as or nearly as small as can be achieved by any estimator. Stated another way, the mle is approximately the MVUE of θ. 31
32 Large Sample Behavior of the MLE Because of this result and the fact that calculus-based techniques can usually be used to derive the mle s (though often numerical methods, such as Newton s method, are necessary), maximum likelihood estimation is the most widely used estimation technique among statisticians. Many of the estimators used in the remainder of the book are mle s. Obtaining an mle, however, does require that the underlying distribution be specified. 32
Point Estimation. Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
6 Point Estimation Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Point Estimation Statistical inference: directed toward conclusions about one or more parameters. We will use the generic
More informationPoint Estimation. Some General Concepts of Point Estimation. Example. Estimator quality
Point Estimation Some General Concepts of Point Estimation Statistical inference = conclusions about parameters Parameters == population characteristics A point estimate of a parameter is a value (based
More informationChapter 6: Point Estimation
Chapter 6: Point Estimation Professor Sharabati Purdue University March 10, 2014 Professor Sharabati (Purdue University) Point Estimation Spring 2014 1 / 37 Chapter Overview Point estimator and point estimate
More informationApplied Statistics I
Applied Statistics I Liang Zhang Department of Mathematics, University of Utah July 14, 2008 Liang Zhang (UofU) Applied Statistics I July 14, 2008 1 / 18 Point Estimation Liang Zhang (UofU) Applied Statistics
More informationLecture 10: Point Estimation
Lecture 10: Point Estimation MSU-STT-351-Sum-17B (P. Vellaisamy: MSU-STT-351-Sum-17B) Probability & Statistics for Engineers 1 / 31 Basic Concepts of Point Estimation A point estimate of a parameter θ,
More informationدرس هفتم یادگیري ماشین. (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی
یادگیري ماشین توزیع هاي نمونه و تخمین نقطه اي پارامترها Sampling Distributions and Point Estimation of Parameter (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی درس هفتم 1 Outline Introduction
More informationCSE 312 Winter Learning From Data: Maximum Likelihood Estimators (MLE)
CSE 312 Winter 2017 Learning From Data: Maximum Likelihood Estimators (MLE) 1 Parameter Estimation Given: independent samples x1, x2,..., xn from a parametric distribution f(x θ) Goal: estimate θ. Not
More informationDefinition 9.1 A point estimate is any function T (X 1,..., X n ) of a random sample. We often write an estimator of the parameter θ as ˆθ.
9 Point estimation 9.1 Rationale behind point estimation When sampling from a population described by a pdf f(x θ) or probability function P [X = x θ] knowledge of θ gives knowledge of the entire population.
More informationBack to estimators...
Back to estimators... So far, we have: Identified estimators for common parameters Discussed the sampling distributions of estimators Introduced ways to judge the goodness of an estimator (bias, MSE, etc.)
More informationPoint Estimators. STATISTICS Lecture no. 10. Department of Econometrics FEM UO Brno office 69a, tel
STATISTICS Lecture no. 10 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 8. 12. 2009 Introduction Suppose that we manufacture lightbulbs and we want to state
More informationLearning From Data: MLE. Maximum Likelihood Estimators
Learning From Data: MLE Maximum Likelihood Estimators 1 Parameter Estimation Assuming sample x1, x2,..., xn is from a parametric distribution f(x θ), estimate θ. E.g.: Given sample HHTTTTTHTHTTTHH of (possibly
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationStatistical estimation
Statistical estimation Statistical modelling: theory and practice Gilles Guillot gigu@dtu.dk September 3, 2013 Gilles Guillot (gigu@dtu.dk) Estimation September 3, 2013 1 / 27 1 Introductory example 2
More informationChapter 4: Commonly Used Distributions. Statistics for Engineers and Scientists Fourth Edition William Navidi
Chapter 4: Commonly Used Distributions Statistics for Engineers and Scientists Fourth Edition William Navidi 2014 by Education. This is proprietary material solely for authorized instructor use. Not authorized
More informationChapter 8. Introduction to Statistical Inference
Chapter 8. Introduction to Statistical Inference Point Estimation Statistical inference is to draw some type of conclusion about one or more parameters(population characteristics). Now you know that a
More informationPoint Estimation. Principle of Unbiased Estimation. When choosing among several different estimators of θ, select one that is unbiased.
Point Estimation Point Estimation Definition A point estimate of a parameter θ is a single number that can be regarded as a sensible value for θ. A point estimate is obtained by selecting a suitable statistic
More information6 Central Limit Theorem. (Chs 6.4, 6.5)
6 Central Limit Theorem (Chs 6.4, 6.5) Motivating Example In the next few weeks, we will be focusing on making statistical inference about the true mean of a population by using sample datasets. Examples?
More informationChapter 7: Estimation Sections
1 / 40 Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: Chapter 7 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods:
More informationChapter 7 - Lecture 1 General concepts and criteria
Chapter 7 - Lecture 1 General concepts and criteria January 29th, 2010 Best estimator Mean Square error Unbiased estimators Example Unbiased estimators not unique Special case MVUE Bootstrap General Question
More informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
More informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 4: Special Discrete Random Variable Distributions Sections 3.7 & 3.8 Geometric, Negative Binomial, Hypergeometric NOTE: The discrete
More informationCIVL Discrete Distributions
CIVL 3103 Discrete Distributions Learning Objectives Define discrete distributions, and identify common distributions applicable to engineering problems. Identify the appropriate distribution (i.e. binomial,
More information6. Genetics examples: Hardy-Weinberg Equilibrium
PBCB 206 (Fall 2006) Instructor: Fei Zou email: fzou@bios.unc.edu office: 3107D McGavran-Greenberg Hall Lecture 4 Topics for Lecture 4 1. Parametric models and estimating parameters from data 2. Method
More informationEngineering Statistics ECIV 2305
Engineering Statistics ECIV 2305 Section 5.3 Approximating Distributions with the Normal Distribution Introduction A very useful property of the normal distribution is that it provides good approximations
More informationINSTITUTE AND FACULTY OF ACTUARIES. Curriculum 2019 SPECIMEN EXAMINATION
INSTITUTE AND FACULTY OF ACTUARIES Curriculum 2019 SPECIMEN EXAMINATION Subject CS1A Actuarial Statistics Time allowed: Three hours and fifteen minutes INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate
More informationDiscrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)
3 Discrete Random Variables and Probability Distributions Stat 4570/5570 Based on Devore s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer
More informationChapter 5: Statistical Inference (in General)
Chapter 5: Statistical Inference (in General) Shiwen Shen University of South Carolina 2016 Fall Section 003 1 / 17 Motivation In chapter 3, we learn the discrete probability distributions, including Bernoulli,
More informationCIVL Learning Objectives. Definitions. Discrete Distributions
CIVL 3103 Discrete Distributions Learning Objectives Define discrete distributions, and identify common distributions applicable to engineering problems. Identify the appropriate distribution (i.e. binomial,
More informationUQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions.
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions. Random Variables 2 A random variable X is a numerical (integer, real, complex, vector etc.) summary of the outcome of the random experiment.
More informationChapter 7: Point Estimation and Sampling Distributions
Chapter 7: Point Estimation and Sampling Distributions Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 20 Motivation In chapter 3, we learned
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More informationStatistics and Their Distributions
Statistics and Their Distributions Deriving Sampling Distributions Example A certain system consists of two identical components. The life time of each component is supposed to have an expentional distribution
More informationSampling and sampling distribution
Sampling and sampling distribution September 12, 2017 STAT 101 Class 5 Slide 1 Outline of Topics 1 Sampling 2 Sampling distribution of a mean 3 Sampling distribution of a proportion STAT 101 Class 5 Slide
More informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 3: Special Discrete Random Variable Distributions Section 3.5 Discrete Uniform Section 3.6 Bernoulli and Binomial Others sections
More informationLecture Notes 6. Assume F belongs to a family of distributions, (e.g. F is Normal), indexed by some parameter θ.
Sufficient Statistics Lecture Notes 6 Sufficiency Data reduction in terms of a particular statistic can be thought of as a partition of the sample space X. Definition T is sufficient for θ if the conditional
More informationMATH 3200 Exam 3 Dr. Syring
. Suppose n eligible voters are polled (randomly sampled) from a population of size N. The poll asks voters whether they support or do not support increasing local taxes to fund public parks. Let M be
More information4-1. Chapter 4. Commonly Used Distributions by The McGraw-Hill Companies, Inc. All rights reserved.
4-1 Chapter 4 Commonly Used Distributions 2014 by The Companies, Inc. All rights reserved. Section 4.1: The Bernoulli Distribution 4-2 We use the Bernoulli distribution when we have an experiment which
More informationStochastic Models. Statistics. Walt Pohl. February 28, Department of Business Administration
Stochastic Models Statistics Walt Pohl Universität Zürich Department of Business Administration February 28, 2013 The Value of Statistics Business people tend to underestimate the value of statistics.
More informationPROBABILITY AND STATISTICS
Monday, January 12, 2015 1 PROBABILITY AND STATISTICS Zhenyu Ye January 12, 2015 Monday, January 12, 2015 2 References Ch10 of Experiments in Modern Physics by Melissinos. Particle Physics Data Group Review
More informationThe Bernoulli distribution
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationEstimating Parameters for Incomplete Data. William White
Estimating Parameters for Incomplete Data William White Insurance Agent Auto Insurance Agency Task Claims in a week 294 340 384 457 680 855 974 1193 1340 1884 2558 9743 Boss, Is this a good representation
More information. (i) What is the probability that X is at most 8.75? =.875
Worksheet 1 Prep-Work (Distributions) 1)Let X be the random variable whose c.d.f. is given below. F X 0 0.3 ( x) 0.5 0.8 1.0 if if if if if x 5 5 x 10 10 x 15 15 x 0 0 x Compute the mean, X. (Hint: First
More informationProbability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 2 Discrete Distributions The binomial distribution 1 Chapter 2 Discrete Distributions Bernoulli trials and the
More informationChapter 5. Statistical inference for Parametric Models
Chapter 5. Statistical inference for Parametric Models Outline Overview Parameter estimation Method of moments How good are method of moments estimates? Interval estimation Statistical Inference for Parametric
More informationM249 Diagnostic Quiz
THE OPEN UNIVERSITY Faculty of Mathematics and Computing M249 Diagnostic Quiz Prepared by the Course Team [Press to begin] c 2005, 2006 The Open University Last Revision Date: May 19, 2006 Version 4.2
More informationChapter 7: Estimation Sections
Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions Frequentist Methods: 7.5 Maximum Likelihood Estimators
More informationPractice Exercises for Midterm Exam ST Statistical Theory - II The ACTUAL exam will consists of less number of problems.
Practice Exercises for Midterm Exam ST 522 - Statistical Theory - II The ACTUAL exam will consists of less number of problems. 1. Suppose X i F ( ) for i = 1,..., n, where F ( ) is a strictly increasing
More informationNon-informative Priors Multiparameter Models
Non-informative Priors Multiparameter Models Statistics 220 Spring 2005 Copyright c 2005 by Mark E. Irwin Prior Types Informative vs Non-informative There has been a desire for a prior distributions that
More informationSome Discrete Distribution Families
Some Discrete Distribution Families ST 370 Many families of discrete distributions have been studied; we shall discuss the ones that are most commonly found in applications. In each family, we need a formula
More information[D7] PROBABILITY DISTRIBUTION OF OUTSTANDING LIABILITY FROM INDIVIDUAL PAYMENTS DATA Contributed by T S Wright
Faculty and Institute of Actuaries Claims Reserving Manual v.2 (09/1997) Section D7 [D7] PROBABILITY DISTRIBUTION OF OUTSTANDING LIABILITY FROM INDIVIDUAL PAYMENTS DATA Contributed by T S Wright 1. Introduction
More informationCHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION
CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Choice Theory Investments 1 / 65 Outline 1 An Introduction
More informationProbability Theory. Probability and Statistics for Data Science CSE594 - Spring 2016
Probability Theory Probability and Statistics for Data Science CSE594 - Spring 2016 What is Probability? 2 What is Probability? Examples outcome of flipping a coin (seminal example) amount of snowfall
More information4-2 Probability Distributions and Probability Density Functions. Figure 4-2 Probability determined from the area under f(x).
4-2 Probability Distributions and Probability Density Functions Figure 4-2 Probability determined from the area under f(x). 4-2 Probability Distributions and Probability Density Functions Definition 4-2
More informationChapter 3 Common Families of Distributions. Definition 3.4.1: A family of pmfs or pdfs is called exponential family if it can be expressed as
Lecture 0 on BST 63: Statistical Theory I Kui Zhang, 09/9/008 Review for the previous lecture Definition: Several continuous distributions, including uniform, gamma, normal, Beta, Cauchy, double exponential
More informationMaximum Likelihood Estimation
Maximum Likelihood Estimation The likelihood and log-likelihood functions are the basis for deriving estimators for parameters, given data. While the shapes of these two functions are different, they have
More informationMuch of what appears here comes from ideas presented in the book:
Chapter 11 Robust statistical methods Much of what appears here comes from ideas presented in the book: Huber, Peter J. (1981), Robust statistics, John Wiley & Sons (New York; Chichester). There are many
More informationChapter 4: Asymptotic Properties of MLE (Part 3)
Chapter 4: Asymptotic Properties of MLE (Part 3) Daniel O. Scharfstein 09/30/13 1 / 1 Breakdown of Assumptions Non-Existence of the MLE Multiple Solutions to Maximization Problem Multiple Solutions to
More informationLecture 5: Fundamentals of Statistical Analysis and Distributions Derived from Normal Distributions
Lecture 5: Fundamentals of Statistical Analysis and Distributions Derived from Normal Distributions ELE 525: Random Processes in Information Systems Hisashi Kobayashi Department of Electrical Engineering
More informationDiscrete Random Variables
Discrete Random Variables ST 370 A random variable is a numerical value associated with the outcome of an experiment. Discrete random variable When we can enumerate the possible values of the variable
More informationFinancial Econometrics
Financial Econometrics Volatility Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) Volatility 01/13 1 / 37 Squared log returns for CRSP daily GPD (TCD) Volatility 01/13 2 / 37 Absolute value
More informationCS 361: Probability & Statistics
March 12, 2018 CS 361: Probability & Statistics Inference Binomial likelihood: Example Suppose we have a coin with an unknown probability of heads. We flip the coin 10 times and observe 2 heads. What can
More informationContinuous random variables
Continuous random variables probability density function (f(x)) the probability distribution function of a continuous random variable (analogous to the probability mass function for a discrete random variable),
More informationStatistics 13 Elementary Statistics
Statistics 13 Elementary Statistics Summer Session I 2012 Lecture Notes 5: Estimation with Confidence intervals 1 Our goal is to estimate the value of an unknown population parameter, such as a population
More informationLattice Model of System Evolution. Outline
Lattice Model of System Evolution Richard de Neufville Professor of Engineering Systems and of Civil and Environmental Engineering MIT Massachusetts Institute of Technology Lattice Model Slide 1 of 48
More informationFocus Points 10/11/2011. The Binomial Probability Distribution and Related Topics. Additional Properties of the Binomial Distribution. Section 5.
The Binomial Probability Distribution and Related Topics 5 Copyright Cengage Learning. All rights reserved. Section 5.3 Additional Properties of the Binomial Distribution Copyright Cengage Learning. All
More informationCase Study: Heavy-Tailed Distribution and Reinsurance Rate-making
Case Study: Heavy-Tailed Distribution and Reinsurance Rate-making May 30, 2016 The purpose of this case study is to give a brief introduction to a heavy-tailed distribution and its distinct behaviors in
More informationLecture 2. Probability Distributions Theophanis Tsandilas
Lecture 2 Probability Distributions Theophanis Tsandilas Comment on measures of dispersion Why do common measures of dispersion (variance and standard deviation) use sums of squares: nx (x i ˆµ) 2 i=1
More informationChapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 1: Introduction Sampling Distributions & the Central Limit Theorem Point Estimation & Estimators Sections 7-1 to 7-2 Sample data
More informationApplications of Good s Generalized Diversity Index. A. J. Baczkowski Department of Statistics, University of Leeds Leeds LS2 9JT, UK
Applications of Good s Generalized Diversity Index A. J. Baczkowski Department of Statistics, University of Leeds Leeds LS2 9JT, UK Internal Report STAT 98/11 September 1998 Applications of Good s Generalized
More informationChapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 29
Chapter 5 Univariate time-series analysis () Chapter 5 Univariate time-series analysis 1 / 29 Time-Series Time-series is a sequence fx 1, x 2,..., x T g or fx t g, t = 1,..., T, where t is an index denoting
More informationReview for Final Exam Spring 2014 Jeremy Orloff and Jonathan Bloom
Review for Final Exam 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom THANK YOU!!!! JON!! PETER!! RUTHI!! ERIKA!! ALL OF YOU!!!! Probability Counting Sets Inclusion-exclusion principle Rule of product
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationConfidence Intervals Introduction
Confidence Intervals Introduction A point estimate provides no information about the precision and reliability of estimation. For example, the sample mean X is a point estimate of the population mean μ
More information1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by. Cov(X, Y ) = E(X E(X))(Y E(Y ))
Correlation & Estimation - Class 7 January 28, 2014 Debdeep Pati Association between two variables 1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by Cov(X, Y ) = E(X E(X))(Y
More informationCharacterization of the Optimum
ECO 317 Economics of Uncertainty Fall Term 2009 Notes for lectures 5. Portfolio Allocation with One Riskless, One Risky Asset Characterization of the Optimum Consider a risk-averse, expected-utility-maximizing
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 207 Homework 5 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 3., Exercises 3, 0. Section 3.3, Exercises 2, 3, 0,.
More informationMVE051/MSG Lecture 7
MVE051/MSG810 2017 Lecture 7 Petter Mostad Chalmers November 20, 2017 The purpose of collecting and analyzing data Purpose: To build and select models for parts of the real world (which can be used for
More informationChapter 5. Sampling Distributions
Lecture notes, Lang Wu, UBC 1 Chapter 5. Sampling Distributions 5.1. Introduction In statistical inference, we attempt to estimate an unknown population characteristic, such as the population mean, µ,
More informationBivariate Birnbaum-Saunders Distribution
Department of Mathematics & Statistics Indian Institute of Technology Kanpur January 2nd. 2013 Outline 1 Collaborators 2 3 Birnbaum-Saunders Distribution: Introduction & Properties 4 5 Outline 1 Collaborators
More informationConjugate priors: Beta and normal Class 15, Jeremy Orloff and Jonathan Bloom
1 Learning Goals Conjugate s: Beta and normal Class 15, 18.05 Jeremy Orloff and Jonathan Bloom 1. Understand the benefits of conjugate s.. Be able to update a beta given a Bernoulli, binomial, or geometric
More informationHardy Weinberg Model- 6 Genotypes
Hardy Weinberg Model- 6 Genotypes Silvelyn Zwanzig Hardy -Weinberg with six genotypes. In a large population of plants (Mimulus guttatus there are possible alleles S, I, F at one locus resulting in six
More informationProbability Distributions for Discrete RV
Probability Distributions for Discrete RV Probability Distributions for Discrete RV Definition The probability distribution or probability mass function (pmf) of a discrete rv is defined for every number
More informationIntroduction to Probability and Inference HSSP Summer 2017, Instructor: Alexandra Ding July 19, 2017
Introduction to Probability and Inference HSSP Summer 2017, Instructor: Alexandra Ding July 19, 2017 Please fill out the attendance sheet! Suggestions Box: Feedback and suggestions are important to the
More information5.3 Statistics and Their Distributions
Chapter 5 Joint Probability Distributions and Random Samples Instructor: Lingsong Zhang 1 Statistics and Their Distributions 5.3 Statistics and Their Distributions Statistics and Their Distributions Consider
More informationEstimation. Focus Points 10/11/2011. Estimating p in the Binomial Distribution. Section 7.3
Estimation 7 Copyright Cengage Learning. All rights reserved. Section 7.3 Estimating p in the Binomial Distribution Copyright Cengage Learning. All rights reserved. Focus Points Compute the maximal length
More informationComputer Statistics with R
MAREK GAGOLEWSKI KONSTANCJA BOBECKA-WESO LOWSKA PRZEMYS LAW GRZEGORZEWSKI Computer Statistics with R 5. Point Estimation Faculty of Mathematics and Information Science Warsaw University of Technology []
More information8.1 Estimation of the Mean and Proportion
8.1 Estimation of the Mean and Proportion Statistical inference enables us to make judgments about a population on the basis of sample information. The mean, standard deviation, and proportions of a population
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 7 Estimation: Single Population Copyright 010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 7-1 Confidence Intervals Contents of this chapter: Confidence
More informationChapter 6 Analyzing Accumulated Change: Integrals in Action
Chapter 6 Analyzing Accumulated Change: Integrals in Action 6. Streams in Business and Biology You will find Excel very helpful when dealing with streams that are accumulated over finite intervals. Finding
More informationTwo Hours. Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER. 22 January :00 16:00
Two Hours MATH38191 Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER STATISTICAL MODELLING IN FINANCE 22 January 2015 14:00 16:00 Answer ALL TWO questions
More informationChapter Learning Objectives. Discrete Random Variables. Chapter 3: Discrete Random Variables and Probability Distributions.
Chapter 3: Discrete Random Variables and Probability Distributions 3-1Discrete Random Variables ibl 3-2 Probability Distributions and Probability Mass Functions 3-33 Cumulative Distribution ib ti Functions
More informationChapter 8: Sampling distributions of estimators Sections
Chapter 8: Sampling distributions of estimators Sections 8.1 Sampling distribution of a statistic 8.2 The Chi-square distributions 8.3 Joint Distribution of the sample mean and sample variance Skip: p.
More informationCentral Limit Theorem (cont d) 7/28/2006
Central Limit Theorem (cont d) 7/28/2006 Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b(n, p, j) we have lim npq b(n, p, np + x npq ) = φ(x), n where φ(x) is
More information9 Expectation and Variance
9 Expectation and Variance Two numbers are often used to summarize a probability distribution for a random variable X. The mean is a measure of the center or middle of the probability distribution, and
More informationLikelihood Methods of Inference. Toss coin 6 times and get Heads twice.
Methods of Inference Toss coin 6 times and get Heads twice. p is probability of getting H. Probability of getting exactly 2 heads is 15p 2 (1 p) 4 This function of p, is likelihood function. Definition:
More informationSTA258H5. Al Nosedal and Alison Weir. Winter Al Nosedal and Alison Weir STA258H5 Winter / 41
STA258H5 Al Nosedal and Alison Weir Winter 2017 Al Nosedal and Alison Weir STA258H5 Winter 2017 1 / 41 NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION. Al Nosedal and Alison Weir STA258H5 Winter 2017
More informationPage Points Score Total: 100
Math 1130 Spring 2019 Sample Midterm 2b 2/28/19 Name (Print): Username.#: Lecturer: Rec. Instructor: Rec. Time: This exam contains 10 pages (including this cover page) and 9 problems. Check to see if any
More informationThe Normal Distribution
Will Monroe CS 09 The Normal Distribution Lecture Notes # July 9, 207 Based on a chapter by Chris Piech The single most important random variable type is the normal a.k.a. Gaussian) random variable, parametrized
More informationMidterm 1, Financial Economics February 15, 2010
Midterm 1, Financial Economics February 15, 2010 Name: Email: @illinois.edu All questions must be answered on this test form. Question 1: Let S={s1,,s11} be the set of states. Suppose that at t=0 the state
More information