DOCSLIB.ORG

The Uniform Distribution, Its Two Parameters, Its Sample Space, Its Expected Value, and Its Variance

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Uniform Distribution, Its Two Parameters, Its Sample Space, Its Expected Value, and Its Variance Introduction Continuous RVs Objectives Uniform Distribution Conclusion Today’s Objectives By the end of this lecture, you should be able to STAT200: Introductory Statistics 1 understand the difference between discrete and continuous random variables; §6.0a: The Uniform 2 know the purpose of the probability density function; Distribution 3 be able to prove that the density is not a probability; 4 know the purpose of the cumulative distribution function; 5 be able to calculate probabilities using geometry or the CDF; and 6 understand the Uniform distribution, its two parameters, its sample space, its expected value, and its variance. STAT200: Introductory Statistics 6.0a: The Uniform Distribution 2 Introduction Introduction Continuous RVs Continuous RVs Continuous RVs Continuous RVs Uniform Distribution Near Continuous RVs Uniform Distribution Near Continuous RVs Conclusion Conclusion Continuous RVs Near Continuous RVs Frequently, a discrete random variable is “essentially” continuous Definition (Continuous Random Variables) because the spacing between adjacent values is insignificant when A continuous random variable is a random variable with a sample compared to the useful data range. space consisting of an interval of values. Examples Examples of near-continuous random variables: 1 GPA Examples 2 Annual salary Examples of continuous random variables: 3 Gross domestic product (GDP) 1 Student height 4 GDP per capita 2 Age of car 5 Crime rate 3 Time spent at a stop light 6 Number of ears of corn grown in Iowa 4 Distance a golf ball goes 7 Number of flu cases in a week STAT200: Introductory Statistics 6.0a: The Uniform Distribution 3 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 4 Introduction Characteristics of the Uniform Introduction Characteristics of the Uniform Continuous RVs Probability Density Function Continuous RVs Probability Density Function Uniform Distribution Example Uniform Distribution Example Conclusion Cumulative Distribution Function Conclusion Cumulative Distribution Function Quantile Function Quantile Function Characteristics of the Uniform Probability Density Function The probability density function for the Uniform distribution is given Definition (Uniform Distribution) by 1 x ∈ [a, b] The Uniform distribution is a continuous distribution that b − a describes random variables whose likelihood of occurring is constant f(x; a, b) = across a specified interval. 0 Otherwise If the random variable X has a Uniform distribution, we write: X s U(a, b) It is characterized by these features Two parameters: minimum and maximum values (a and b) Sample space is S = [a, b] E a+b X = 2 V (b−a)2 X = 12 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 5 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 6 Introduction Characteristics of the Uniform Introduction Characteristics of the Uniform Continuous RVs Probability Density Function Continuous RVs Probability Density Function Uniform Distribution Example Uniform Distribution Example Conclusion Cumulative Distribution Function Conclusion Cumulative Distribution Function Quantile Function Quantile Function Purposes of pdf Example I: Main and Academy Example Note that the probability density function has two purposes: 1 To help the researcher understand probability for a continuous There is only one stop light between home and school in the morning. distribution It regularly cycles among green (175s), yellow (5s), and red (180s). 2 To help the researcher calculate probabilities of a continuous Given that I stop at the light, what is the probability that I wait at random variable most 60 seconds? Important!! Solution: Because the time I wait does not depend on when I get there, and because there is a definite lower- and upper-bound to my wait time (0 Probabilities are areas under the density curve. and 180s), the wait time distribution follows a Uniform distribution. If we define T as the time (in seconds) I spend waiting at this light, then T s U(0, 180) We are asked to calculate P[T ≤ 60]. STAT200: Introductory Statistics 6.0a: The Uniform Distribution 7 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 8 Introduction Characteristics of the Uniform Introduction Characteristics of the Uniform Continuous RVs Probability Density Function Continuous RVs Probability Density Function Uniform Distribution Example Uniform Distribution Example Conclusion Cumulative Distribution Function Conclusion Cumulative Distribution Function Quantile Function Quantile Function Example I: Main and Academy Example I: Main and Academy Because the pdf of the Uniform distribution is just a rectangle, and Graphically, this is what we just described because areas in pdfs are probabilities, we just need to calculate the area of the region T ≤ 60. This is just a rectangle. Its area is its height times its width. From geometry: 1 Height = 180 Width = 60 1 Area (Probability) = 180 × 60 = 0.33333... Thus, the probability of waiting at most 60s is 33.3%. Doing this in R just requires the following line: punif(60, min=0, max=180) STAT200: Introductory Statistics 6.0a: The Uniform Distribution 9 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 10 Introduction Characteristics of the Uniform Introduction Characteristics of the Uniform Continuous RVs Probability Density Function Continuous RVs Probability Density Function Uniform Distribution Example Uniform Distribution Example Conclusion Cumulative Distribution Function Conclusion Cumulative Distribution Function Quantile Function Quantile Function Cumulative Distribution Function Uniform CDF The cumulative distribution function (CDF) is the function that describes F (x) = P[X ≤ x]. This can be easily calculated for the Uniform distribution, in general. Definition (Cumulative Distribution Function) The cumulative distribution function (CDF) of a probability distribution is defined as F (x) = P[X ≤ x] This is frequently difficult to calculate. Usually, it requires integral calculus. However, for the Uniform distribution, we can rely on middle-school geometry. Remember: Probabilities are just areas under the pdf curve. STAT200: Introductory Statistics 6.0a: The Uniform Distribution 11 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 12 Introduction Characteristics of the Uniform Introduction Characteristics of the Uniform Continuous RVs Probability Density Function Continuous RVs Probability Density Function Uniform Distribution Example Uniform Distribution Example Conclusion Cumulative Distribution Function Conclusion Cumulative Distribution Function Quantile Function Quantile Function Cumulative Distribution Function Quantile Function The CDF starts with a value of the random variable, x, and calculates The cumulative distribution function (CDF) is the function that a cumulative probability, p, such that P[X ≤ x] = p. The quantile describes F (x) = P[X ≤ x]. This can be easily calculated for the function starts with a cumulative probability, p, and calculates the Uniform distribution, in general. Let x-value such that P[X ≤ x] = p. X s U(a, b) The two functions are inverses of each other. That is, Q(p) = F −1(p). Thus, the quantile function for the Uniform P The probability [X ≤ x] is the area of the rectangle with distribution is height f(x) = 1 b−a x − a p = width: x − a b − a p(b − a) = x − a This means the cumulative distribution function is p(b − a) + a = x x − a P[X ≤ x] = F (x) = That is, the quantile function for the Uniform distribution is b − a Q(p) = p(b − a) + a STAT200: Introductory Statistics 6.0a: The Uniform Distribution 13 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 14 Introduction Summary Introduction Summary Continuous RVs The Future Continuous RVs The Future Uniform Distribution R Functions Uniform Distribution R Functions Conclusion Some Readings Conclusion Some Readings Calculus Extra Calculus Extra Uniform Summary The Future Here is what we learned in this slide deck: Continuous random variables describe different phenomena than discrete random variables In the future, we will Instead of a probability mass function (discrete), there is a examine the Exponential distribution probability density function (pdf), which describes a probabilty density, not a probability examine the Normal distribution Probability is the area under the pdf curve practice calculating probabilities using formulas, tables, and R The cumulative distribution function (CDF) is the probability continue thinking about the relationship between the random P[X ≤ x] variables around us and their distributions The Uniform distribution describes a random variable where all values are equally likely a+b The mean of a Uniform random variable is 2 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 15 STAT200: Introductory Statistics 6.0a: The Uniform Distribution 16 Introduction Summary Introduction Summary Continuous RVs The Future Continuous RVs The Future Uniform Distribution R Functions Uniform Distribution R Functions Conclusion Some Readings Conclusion Some Readings Calculus Extra Calculus Extra R Functions Readings In this slide deck, we covered two R functions. This is in addition to one we have already experienced and one we will experience: dunif(x, min,max) is the density function, f(x) Course Readings: Hawkes Learning: None P punif(x, min,max) = F (x) = [X ≤ x] R for Starters: Appendix B.1, B.2 runif(n, min,max) gives a random sample from this Supplementary Readings distribution (the random function) Wikipedia: Uniform Distribution (Continuous)
Load more

Recommended publications
  • 5. the Student T Distribution
    Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way.
    [Show full text]
  • A Tail Quantile Approximation Formula for the Student T and the Symmetric Generalized Hyperbolic Distribution
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Schlüter, Stephan; Fischer, Matthias J. Working Paper A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution IWQW Discussion Papers, No. 05/2009 Provided in Cooperation with: Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics Suggested Citation: Schlüter, Stephan; Fischer, Matthias J. (2009) : A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution, IWQW Discussion Papers, No. 05/2009, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung (IWQW), Nürnberg This Version is available at: http://hdl.handle.net/10419/29554 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu IWQW Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung Diskussionspapier Discussion Papers No.
    [Show full text]
  • Sampling Student's T Distribution – Use of the Inverse Cumulative
    Sampling Student’s T distribution – use of the inverse cumulative distribution function William T. Shaw Department of Mathematics, King’s College, The Strand, London WC2R 2LS, UK With the current interest in copula methods, and fat-tailed or other non-normal distributions, it is appropriate to investigate technologies for managing marginal distributions of interest. We explore “Student’s” T distribution, survey its simulation, and present some new techniques for simulation. In particular, for a given real (not necessarily integer) value n of the number of degrees of freedom, −1 we give a pair of power series approximations for the inverse, Fn ,ofthe cumulative distribution function (CDF), Fn.Wealsogivesomesimpleandvery fast exact and iterative techniques for defining this function when n is an even −1 integer, based on the observation that for such cases the calculation of Fn amounts to the solution of a reduced-form polynomial equation of degree n − 1. We also explain the use of Cornish–Fisher expansions to define the inverse CDF as the composition of the inverse CDF for the normal case with a simple polynomial map. The methods presented are well adapted for use with copula and quasi-Monte-Carlo techniques. 1 Introduction There is much interest in many areas of financial modeling on the use of copulas to glue together marginal univariate distributions where there is no easy canonical multivariate distribution, or one wishes to have flexibility in the mechanism for combination. One of the more interesting marginal distributions is the “Student’s” T distribution. This statistical distribution was published by W. Gosset in 1908.
    [Show full text]
  • Nonparametric Multivariate Kurtosis and Tailweight Measures
    Nonparametric Multivariate Kurtosis and Tailweight Measures Jin Wang1 Northern Arizona University and Robert Serfling2 University of Texas at Dallas November 2004 – final preprint version, to appear in Journal of Nonparametric Statistics, 2005 1Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, Arizona 86011-5717, USA. Email: [email protected]. 2Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75083- 0688, USA. Email: [email protected]. Website: www.utdallas.edu/∼serfling. Support by NSF Grant DMS-0103698 is gratefully acknowledged. Abstract For nonparametric exploration or description of a distribution, the treatment of location, spread, symmetry and skewness is followed by characterization of kurtosis. Classical moment- based kurtosis measures the dispersion of a distribution about its “shoulders”. Here we con- sider quantile-based kurtosis measures. These are robust, are defined more widely, and dis- criminate better among shapes. A univariate quantile-based kurtosis measure of Groeneveld and Meeden (1984) is extended to the multivariate case by representing it as a transform of a dispersion functional. A family of such kurtosis measures defined for a given distribution and taken together comprises a real-valued “kurtosis functional”, which has intuitive appeal as a convenient two-dimensional curve for description of the kurtosis of the distribution. Several multivariate distributions in any dimension may thus be compared with respect to their kurtosis in a single two-dimensional plot. Important properties of the new multivariate kurtosis measures are established. For example, for elliptically symmetric distributions, this measure determines the distribution within affine equivalence. Related tailweight measures, influence curves, and asymptotic behavior of sample versions are also discussed.
    [Show full text]
  • Variance Reduction Techniques for Estimating Quantiles and Value-At-Risk" (2010)
    New Jersey Institute of Technology Digital Commons @ NJIT Dissertations Electronic Theses and Dissertations Spring 5-31-2010 Variance reduction techniques for estimating quantiles and value- at-risk Fang Chu New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/dissertations Part of the Databases and Information Systems Commons, and the Management Information Systems Commons Recommended Citation Chu, Fang, "Variance reduction techniques for estimating quantiles and value-at-risk" (2010). Dissertations. 212. https://digitalcommons.njit.edu/dissertations/212 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Dissertations by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Cprht Wrnn & trtn h prht l f th Untd Stt (tl , Untd Stt Cd vrn th n f phtp r thr rprdtn f prhtd trl. Undr rtn ndtn pfd n th l, lbrr nd rhv r thrzd t frnh phtp r thr rprdtn. On f th pfd ndtn tht th phtp r rprdtn nt t b “d fr n prp thr thn prvt td, hlrhp, r rrh. If , r rt fr, r ltr , phtp r rprdtn fr prp n x f “fr tht r b lbl fr prht nfrnnt, h ntttn rrv th rht t rf t pt pn rdr f, n t jdnt, flfllnt f th rdr ld nvlv vltn f prht l. l t: h thr rtn th prht hl th r Inttt f hnl rrv th rht t dtrbt th th r drttn rntn nt: If d nt h t prnt th p, thn lt “ fr: frt p t: lt p n th prnt dl rn h n tn lbrr h rvd f th prnl nfrtn nd ll ntr fr th pprvl p nd brphl th f th nd drttn n rdr t prtt th dntt f I rdt nd flt.
    [Show full text]
  • Modified Odd Weibull Family of Distributions: Properties and Applications Christophe Chesneau, Taoufik El Achi
    Modified Odd Weibull Family of Distributions: Properties and Applications Christophe Chesneau, Taoufik El Achi To cite this version: Christophe Chesneau, Taoufik El Achi. Modified Odd Weibull Family of Distributions: Properties and Applications. 2019. hal-02317235 HAL Id: hal-02317235 https://hal.archives-ouvertes.fr/hal-02317235 Submitted on 15 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Modified Odd Weibull Family of Distributions: Properties and Applications Christophe CHESNEAU and Taoufik EL ACHI LMNO, University of Caen Normandie, 14000, Caen, France 17 August 2019 Abstract In this paper, a new family of continuous distributions, called the modified odd Weibull-G (MOW-G) family, is studied. The MOW-G family has the feature to use the Weibull distribu- tion as main generator and a new modification of the odd transformation, opening new horizon in terms of statistical modelling. Its main theoretical and practical aspects are explored. In particular, for the mathematical properties, we investigate some results in distribution, quantile function, skewness, kurtosis, moments, moment generating function, order statistics and en- tropy. For the statistical aspect, the maximum likelihood estimation method is used to estimate the model parameters.
    [Show full text]
  • Quantile Function Methods for Decision Analysis
    QUANTILE FUNCTION METHODS FOR DECISION ANALYSIS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MANAGEMENT SCIENCE AND ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Bradford W. Powley August 2013 © 2013 by Bradford William Powley. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/yn842pf8910 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ronald Howard, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ross Shachter I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Tom Keelin Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract Encoding prior probability distributions is a fundamental step in any decision analysis. A decision analyst often elicits an expert's knowledge about a continuous uncertain quantity as a set of quantile-probability pairs (points on a cumulative distribution function) and seeks a probability distribution consistent with them.
    [Show full text]
  • Linear Moments: an Overview
    International Journal of Business and Statistical Analysis ISSN (2384-4663) Int. J. Bus. Stat. Ana. 2, No. 2 (July-2015) Linear Moments: An Overview A.F. Kandeel1 1 Department of Statistics and Mathematics, Faculty of Commerece, Benha University, Benha, Egypt Received 5 February 2015, Revised 1 April,. 2015, Accepted 5 April 2015, Published 1st July 2015 Abstract: Many statistical techniques are based on the use of linear combinations of order statistics that called linear moments. L-moments are a sequence of statistics used to summarize the shape of a probability distribution. They are linear combinations of order statistics analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively. In this paper an overview for recent works in L- moments is presented. Keywords: Order statistics, L-moments, TL-moments, LQ-moments, Quantile function. to the standard errors of L-moments to be finite, only the 1. INTRODUCTION distribution is required to have finite variance and no Many statistical techniques are based on the use of higher-order moments need be finite. Also, though linear combinations of order statistics (or quantile moment ratios can be arbitrarily large, sample moment function), but there has not been developed a unified ratios have algebraic bounds but sample L-moment ratios theory of estimation covering the characterization of can take any values that the corresponding population probability distribution, until Hosking (1990) introduced quantities can. L-moments, as an alternative to the classical moments. In addition, L-moments have properties that hold in a So, L-moments are summary statistics for probability wide range of practical situations.
    [Show full text]
  • Quantile Mechanics: Issues Arising from Critical Review
    International Journal of Advanced and Applied Sciences, 6(1) 2019, Pages: 9-23 Contents lists available at Science-Gate International Journal of Advanced and Applied Sciences Journal homepage: http://www.science-gate.com/IJAAS.html Quantile mechanics: Issues arising from critical review Hilary I. Okagbue 1, *, Muminu O. Adamu 2, Timothy A. Anake 1 1Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria 2Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria ARTICLE INFO ABSTRACT Article history: Approximations are the alternative way of obtaining the Quantile function Received 26 June 2018 when the inversion method cannot be applied to distributions whose Received in revised form cumulative distribution functions do not have close form expressions. 27 October 2018 Approximations come in form of functional approximation, numerical Accepted 27 October 2018 algorithm, closed form expressed in terms of others and series expansions. Several quantile approximations are available which have been proven to be Keywords: precise, but some issues like the presence of shape parameters, Approximation inapplicability of existing methods to complex distributions and low Cumulative distribution function computational speed and accuracy place undue limitations to their effective Probability density function use. Quantile mechanics (QM) is a series expansion method that addressed Quantile function these issues as evidenced in the paper. Quantile mechanics is a generalization Quantile mechanics of the use of ordinary differential equations (ODE) in quantile approximation. Series expansion The paper is a review that critically examined with evidences; the formulation, applications and advantages of QM over other surveyed methods. Some issues bothering on the use of QM were also discussed. The review concluded with areas of further studies which are open for scientific investigation and exploration.
    [Show full text]
  • Statistics for Data Science
    Statistics for Data Science MSc Data Science WiSe 2019/20 Prof. Dr. Dirk Ostwald 1 (2) Random variables 2 Random variables • Definition and notation • Cumulative distribution functions • Probability mass and density functions 3 Random variables • Definition and notation • Cumulative distribution functions • Probability mass and density functions 4 Definition and notation Random variables and distributions • Let (Ω; A; P) be a probability space and let X :Ω !X be a function. • Let S be a σ-algebra on X . • For every S 2 S let the preimage of S be X−1(S) := f! 2 ΩjX(!) 2 Sg: (1) • If X−1(S) 2 A for all S 2 S, then X is called measurable. • Let X :Ω !X be measurable. All S 2 S get allocated the probability −1 PX : S! [0; 1];S 7! PX (S) := P X (S) = P (f! 2 ΩjX(!) 2 Sg) (2) • X is called a random variable and PX is called the distribution of X. • (X ; S; PX ) is a probability space. • With X = R and S = B the probability space (R; B; PX ) takes center stage. 5 Definition and notation Random variables and distributions 6 Definition and notation Definition (Random variable) Let (Ω; A; P) denote a probability space. A (real-valued) random variable is a mapping X :Ω ! R;! 7! X(!); (3) with the measurability property f! 2 ΩjX(!) 2 S)g 2 A for all S 2 S: (4) Remarks • Random variables are neither \random" nor \variables". • Intuitively, ! 2 Ω gets randomly selected according to P and X(!) realized. • The distributions (probability measures) of random variables are central.
    [Show full text]
  • MTM Estimators for the Parameters of T and Gamma Distributions
    Under consideration for publication in Euro. Jnl of Applied Mathematics 1 A Statistical Application of the Quantile Mechanics Approach: MTM Estimators for the Parameters of t and Gamma Distributions A. KLEEFELD 1 and V. BRAZAUSKAS 2 1 Brandenburgische Technische Universit¨at, Department I (Mathematics, Natural Sciences, Computer Science), P.O. Box 101344, 03013 Cottbus, Germany. email: [email protected] 2 Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201, USA. email: [email protected] (Received 18 January 2011) In this article, we revisit the quantile mechanics approach which was introduced by Stein- brecher and Shaw [18]. Our objectives are: (i) to derive the method of trimmed moments (MTM) estimators for the parameters of gamma and Student’s t distributions, and (ii) to examine their large- and small-sample statistical properties. Since trimmed moments are defined through the quantile function of the distribution, quantile mechanics seems like a natural approach for achieving the objective (i). To accomplish the second goal, we rely on the general large-sample results for MTMs, which were established by Brazauskas, Jones, and Zitikis [3], and then use Monte Carlo simulations to investigate small-sample behaviour of the newly derived estimators. We find that, unlike the maximum likelihood method which usually yields fully efficient but non-robust estimators, the MTM estima- tors are robust and offer competitive trade-offs between robustness and efficiency. These properties are essential when one employs gamma or Student’s t distributions in such outlier-prone areas as insurance and finance. Key Words: Point estimation (62F10), Asymptotic properties of estimators (62F12), Robust- ness and adaptive procedures (62F35), Method of trimmed moments (62F99), Computational problems in statistics (65C60).
    [Show full text]
  • The Log-Odd Normal Generalized Family of Distributions with Application
    Anais da Academia Brasileira de Ciências 2019 91(2): e20180207 (Annals of the Brazilian Academy of Sciences) Printed version ISSN 0001-3765 / Online version ISSN 1678-2690 http://dx.doi.org/10.1590/0001-3765201920180207 www.scielo.br/aabc | www.fb.com/aabcjournal The Log-Odd Normal Generalized Family of Distributions with Application MUHAMMAD ZUBAIR1, TIBOR K. POGÁNY2, GAUSS M. CORDEIRO3 and MUHAMMAD H. TAHIR4 1Department of Statistics, Government S.E. College, Bahawalpur, 63100, Pakistan 2Faculty of Maritime Studies, University of Rijeka, HR-51000 Rijeka, Croatia & Applied Mathematics Institute, Óbuda University, H-1034 Budapest, Hungary 3Departamento de Estatística, Universidade Federal de Pernambuco, Av. Prof. Moraes Rego, 1235, Cidade Universitária, 50740-540 Recife, PE, Brazil 4Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan Manuscript received on March 20, 2018; accepted for publication on July 17, 2018 How to cite: ZUBAIR M, POGÁNY TK, GAUSS MC AND TAHIR MH. 2019. The Log-Odd Normal Generalized Family of Distributions with Application. An Acad Bras Cienc 91: e20180207. DOI 10.1590/10.1590/0001-3765201920180207. Abstract: The normal distribution has a central place in distribution theory and statistics. We propose the log-odd normal generalized (LONG) family of distributions based on log-odds and obtain some of its mathematical properties including a useful linear representation for the new family. We investigate, as a special model, the log-odd normal power-Cauchy (LONPC) distribution. Some structural properties of LONPC distribution are obtained including quantile function, ordinary and incomplete moments, generating function and some asymptotics. We estimate the model parameters using the maximum likelihood method.
    [Show full text]