Why Heavy Tails?
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Circularly-Symmetric Complex Normal Ratio Distribution for Scalar Transmissibility Functions. Part II: Probabilistic Model and Validation
Mechanical Systems and Signal Processing 80 (2016) 78–98 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp Circularly-symmetric complex normal ratio distribution for scalar transmissibility functions. Part II: Probabilistic model and validation Wang-Ji Yan, Wei-Xin Ren n Department of Civil Engineering, Hefei University of Technology, Hefei, Anhui 23009, China article info abstract Article history: In Part I of this study, some new theorems, corollaries and lemmas on circularly- Received 29 May 2015 symmetric complex normal ratio distribution have been mathematically proved. This Received in revised form part II paper is dedicated to providing a rigorous treatment of statistical properties of raw 3 February 2016 scalar transmissibility functions at an arbitrary frequency line. On the basis of statistics of Accepted 27 February 2016 raw FFT coefficients and circularly-symmetric complex normal ratio distribution, explicit Available online 23 March 2016 closed-form probabilistic models are established for both multivariate and univariate Keywords: scalar transmissibility functions. Also, remarks on the independence of transmissibility Transmissibility functions at different frequency lines and the shape of the probability density function Uncertainty quantification (PDF) of univariate case are presented. The statistical structures of probabilistic models are Signal processing concise, compact and easy-implemented with a low computational effort. They hold for Statistics Structural dynamics general stationary vector processes, either Gaussian stochastic processes or non-Gaussian Ambient vibration stochastic processes. The accuracy of proposed models is verified using numerical example as well as field test data of a high-rise building and a long-span cable-stayed bridge. -
Maria Gaetana Agnesi Melissa De La Cruz
Maria Gaetana Agnesi Melissa De La Cruz Maria Gaetana Agnesi's life Maria Gaetana Agnesi was born on May 16, 1718. She also died on January 9, 1799. She was an Italian mathematician, philosopher, theologian, and humanitarian. Her father, Pietro Agnesi, was either said to be a wealthy silk merchant or a mathematics professor. Regardless, he was well to do and lived a social climber. He married Anna Fortunata Brivio, but had a total of 21 children with 3 female relations. He used his kids to be known with the Milanese socialites of the time and Maria Gaetana Agnesi was his oldest. He hosted soires and used his kids as entertainment. Maria's sisters would perform the musical acts while she would present Latin debates and orations on questions of natural philosophy. The family was well to do and Pietro has to make sure his children were under the best training to entertain well at these parties. For Maria, that meant well-esteemed tutors would be given because her father discovered her great potential for an advanced intellect. Her education included the usual languages and arts; she spoke Italian and French at the age of 5 and but the age of 11, she also knew how to speak Greek, Hebrew, Spanish, German and Latin. Historians have given her the name of the Seven-Tongued Orator. With private tutoring, she was also taught topics such as mathematics and natural philosophy, which was unusual for women at that time. With the Agnesi family living in Milan, Italy, Maria's family practiced religion seriously. -
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. -
A Study of Non-Central Skew T Distributions and Their Applications in Data Analysis and Change Point Detection
A STUDY OF NON-CENTRAL SKEW T DISTRIBUTIONS AND THEIR APPLICATIONS IN DATA ANALYSIS AND CHANGE POINT DETECTION Abeer M. Hasan A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Arjun K. Gupta, Co-advisor Wei Ning, Advisor Mark Earley, Graduate Faculty Representative Junfeng Shang. Copyright c August 2013 Abeer M. Hasan All rights reserved iii ABSTRACT Arjun K. Gupta, Co-advisor Wei Ning, Advisor Over the past three decades there has been a growing interest in searching for distribution families that are suitable to analyze skewed data with excess kurtosis. The search started by numerous papers on the skew normal distribution. Multivariate t distributions started to catch attention shortly after the development of the multivariate skew normal distribution. Many researchers proposed alternative methods to generalize the univariate t distribution to the multivariate case. Recently, skew t distribution started to become popular in research. Skew t distributions provide more flexibility and better ability to accommodate long-tailed data than skew normal distributions. In this dissertation, a new non-central skew t distribution is studied and its theoretical properties are explored. Applications of the proposed non-central skew t distribution in data analysis and model comparisons are studied. An extension of our distribution to the multivariate case is presented and properties of the multivariate non-central skew t distri- bution are discussed. We also discuss the distribution of quadratic forms of the non-central skew t distribution. In the last chapter, the change point problem of the non-central skew t distribution is discussed under different settings. -
Basic Econometrics / Statistics Statistical Distributions: Normal, T, Chi-Sq, & F
Basic Econometrics / Statistics Statistical Distributions: Normal, T, Chi-Sq, & F Course : Basic Econometrics : HC43 / Statistics B.A. Hons Economics, Semester IV/ Semester III Delhi University Course Instructor: Siddharth Rathore Assistant Professor Economics Department, Gargi College Siddharth Rathore guj75845_appC.qxd 4/16/09 12:41 PM Page 461 APPENDIX C SOME IMPORTANT PROBABILITY DISTRIBUTIONS In Appendix B we noted that a random variable (r.v.) can be described by a few characteristics, or moments, of its probability function (PDF or PMF), such as the expected value and variance. This, however, presumes that we know the PDF of that r.v., which is a tall order since there are all kinds of random variables. In practice, however, some random variables occur so frequently that statisticians have determined their PDFs and documented their properties. For our purpose, we will consider only those PDFs that are of direct interest to us. But keep in mind that there are several other PDFs that statisticians have studied which can be found in any standard statistics textbook. In this appendix we will discuss the following four probability distributions: 1. The normal distribution 2. The t distribution 3. The chi-square (2 ) distribution 4. The F distribution These probability distributions are important in their own right, but for our purposes they are especially important because they help us to find out the probability distributions of estimators (or statistics), such as the sample mean and sample variance. Recall that estimators are random variables. Equipped with that knowledge, we will be able to draw inferences about their true population values. -
Beta-Cauchy Distribution: Some Properties and Applications
Journal of Statistical Theory and Applications, Vol. 12, No. 4 (December 2013), 378-391 Beta-Cauchy Distribution: Some Properties and Applications Etaf Alshawarbeh Department of Mathematics, Central Michigan University Mount Pleasant, MI 48859, USA Email: [email protected] Felix Famoye Department of Mathematics, Central Michigan University Mount Pleasant, MI 48859, USA Email: [email protected] Carl Lee Department of Mathematics, Central Michigan University Mount Pleasant, MI 48859, USA Email: [email protected] Abstract Some properties of the four-parameter beta-Cauchy distribution such as the mean deviation and Shannon’s entropy are obtained. The method of maximum likelihood is proposed to estimate the parameters of the distribution. A simulation study is carried out to assess the performance of the maximum likelihood estimates. The usefulness of the new distribution is illustrated by applying it to three empirical data sets and comparing the results to some existing distributions. The beta-Cauchy distribution is found to provide great flexibility in modeling symmetric and skewed heavy-tailed data sets. Keywords and Phrases: Beta family, mean deviation, entropy, maximum likelihood estimation. 1. Introduction Eugene et al.1 introduced a class of new distributions using beta distribution as the generator and pointed out that this can be viewed as a family of generalized distributions of order statistics. Jones2 studied some general properties of this family. This class of distributions has been referred to as “beta-generated distributions”.3 This family of distributions provides some flexibility in modeling data sets with different shapes. Let F(x) be the cumulative distribution function (CDF) of a random variable X, then the CDF G(x) for the “beta-generated distributions” is given by Fx() 1 11 G( x ) [ B ( , )] t(1 t ) dt , 0 , , (1) 0 where B(αβ , )=ΓΓ ( α ) ( β )/ Γ+ ( α β ) . -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 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. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
Chapter 7 “Continuous Distributions”.Pdf
CHAPTER 7 Continuous distributions 7.1. Basic theory 7.1.1. Denition, PDF, CDF. We start with the denition a continuous random variable. Denition (Continuous random variables) A random variable X is said to have a continuous distribution if there exists a non- negative function f = f such that X ¢ b P(a 6 X 6 b) = f(x)dx a for every a and b. The function f is called the density function for X or the PDF for X. ¡More precisely, such an X is said to have an absolutely continuous¡ distribution. Note that 1 . In particular, a for every . −∞ f(x)dx = P(−∞ < X < 1) = 1 P(X = a) = a f(x)dx = 0 a 3 ¡Example 7.1. Suppose we are given that f(x) = c=x for x > 1 and 0 otherwise. Since 1 and −∞ f(x)dx = 1 ¢ ¢ 1 1 1 c c f(x)dx = c 3 dx = ; −∞ 1 x 2 we have c = 2. PMF or PDF? Probability mass function (PMF) and (probability) density function (PDF) are two names for the same notion in the case of discrete random variables. We say PDF or simply a density function for a general random variable, and we use PMF only for discrete random variables. Denition (Cumulative distribution function (CDF)) The distribution function of X is dened as ¢ y F (y) = FX (y) := P(−∞ < X 6 y) = f(x)dx: −∞ It is also called the cumulative distribution function (CDF) of X. 97 98 7. CONTINUOUS DISTRIBUTIONS We can dene CDF for any random variable, not just continuous ones, by setting F (y) := P(X 6 y). -
(Introduction to Probability at an Advanced Level) - All Lecture Notes
Fall 2018 Statistics 201A (Introduction to Probability at an advanced level) - All Lecture Notes Aditya Guntuboyina August 15, 2020 Contents 0.1 Sample spaces, Events, Probability.................................5 0.2 Conditional Probability and Independence.............................6 0.3 Random Variables..........................................7 1 Random Variables, Expectation and Variance8 1.1 Expectations of Random Variables.................................9 1.2 Variance................................................ 10 2 Independence of Random Variables 11 3 Common Distributions 11 3.1 Ber(p) Distribution......................................... 11 3.2 Bin(n; p) Distribution........................................ 11 3.3 Poisson Distribution......................................... 12 4 Covariance, Correlation and Regression 14 5 Correlation and Regression 16 6 Back to Common Distributions 16 6.1 Geometric Distribution........................................ 16 6.2 Negative Binomial Distribution................................... 17 7 Continuous Distributions 17 7.1 Normal or Gaussian Distribution.................................. 17 1 7.2 Uniform Distribution......................................... 18 7.3 The Exponential Density...................................... 18 7.4 The Gamma Density......................................... 18 8 Variable Transformations 19 9 Distribution Functions and the Quantile Transform 20 10 Joint Densities 22 11 Joint Densities under Transformations 23 11.1 Detour to Convolutions...................................... -
Chapter 10 “Some Continuous Distributions”.Pdf
CHAPTER 10 Some continuous distributions 10.1. Examples of continuous random variables We look at some other continuous random variables besides normals. Uniform distribution A continuous random variable has uniform distribution if its density is f(x) = 1/(b a) − if a 6 x 6 b and 0 otherwise. For a random variable X with uniform distribution its expectation is 1 b a + b EX = x dx = . b a ˆ 2 − a Exponential distribution A continuous random variable has exponential distribution with parameter λ > 0 if its λx density is f(x) = λe− if x > 0 and 0 otherwise. Suppose X is a random variable with an exponential distribution with parameter λ. Then we have ∞ λx λa (10.1.1) P(X > a) = λe− dx = e− , ˆa λa FX (a) = 1 P(X > a) = 1 e− , − − and we can use integration by parts to see that EX = 1/λ, Var X = 1/λ2. Examples where an exponential random variable is a good model is the length of a telephone call, the length of time before someone arrives at a bank, the length of time before a light bulb burns out. Exponentials are memoryless, that is, P(X > s + t X > t) = P(X > s), | or given that the light bulb has burned 5 hours, the probability it will burn 2 more hours is the same as the probability a new light bulb will burn 2 hours. Here is how we can prove this 129 130 10. SOME CONTINUOUS DISTRIBUTIONS P(X > s + t) P(X > s + t X > t) = | P(X > t) λ(s+t) e− λs − = λt = e e− = P(X > s), where we used Equation (10.1.1) for a = t and a = s + t. -
Prerequisites for Calculus
5128_Ch01_pp02-57.qxd 1/13/06 8:47 AM Page 2 Chapter 1 Prerequisites for Calculus xponential functions are used to model situations in which growth or decay change Edramatically. Such situations are found in nuclear power plants, which contain rods of plutonium-239; an extremely toxic radioactive isotope. Operating at full capacity for one year, a 1,000 megawatt power plant discharges about 435 lb of plutonium-239. With a half-life of 24,400 years, how much of the isotope will remain after 1,000 years? This question can be answered with the mathematics covered in Section 1.3. 2 5128_Ch01_pp02-57.qxd 1/13/06 8:47 AM Page 3 Section 1.1 Lines 3 Chapter 1 Overview This chapter reviews the most important things you need to know to start learning calcu- lus. It also introduces the use of a graphing utility as a tool to investigate mathematical ideas, to support analytic work, and to solve problems with numerical and graphical meth- ods. The emphasis is on functions and graphs, the main building blocks of calculus. Functions and parametric equations are the major tools for describing the real world in mathematical terms, from temperature variations to planetary motions, from brain waves to business cycles, and from heartbeat patterns to population growth. Many functions have particular importance because of the behavior they describe. Trigonometric func- tions describe cyclic, repetitive activity; exponential, logarithmic, and logistic functions describe growth and decay; and polynomial functions can approximate these and most other functions. 1.1 Lines What you’ll learn about Increments • Increments One reason calculus has proved to be so useful is that it is the right mathematics for relat- • Slope of a Line ing the rate of change of a quantity to the graph of the quantity. -
Distributions (3) © 2008 Winton 2 VIII
© 2008 Winton 1 Distributions (3) © onntiW8 020 2 VIII. Lognormal Distribution • Data points t are said to be lognormally distributed if the natural logarithms, ln(t), of these points are normally distributed with mean μ ion and standard deviatσ – If the normal distribution is sampled to get points rsample, then the points ersample constitute sample values from the lognormal distribution • The pdf or the lognormal distribution is given byf - ln(x)()μ 2 - 1 2 f(x)= e ⋅ 2 σ >(x 0) σx 2 π because ∞ - ln(x)()μ 2 - ∞ -() t μ-2 1 1 2 1 2 ∫ e⋅ 2eσ dx = ∫ dt ⋅ 2σ (where= t ln(x)) 0 xσ2 π 0σ2 π is the pdf for the normal distribution © 2008 Winton 3 Mean and Variance for Lognormal • It can be shown that in terms of μ and σ σ2 μ + E(X)= 2 e 2⋅μ σ + 2 σ2 Var(X)= e( ⋅ ) e - 1 • If the mean E and variance V for the lognormal distribution are given, then the corresponding μ and σ2 for the normal distribution are given by – σ2 = log(1+V/E2) – μ = log(E) - σ2/2 • The lognormal distribution has been used in reliability models for time until failure and for stock price distributions – The shape is similar to that of the Gamma distribution and the Weibull distribution for the case α > 2, but the peak is less towards 0 © 2008 Winton 4 Graph of Lognormal pdf f(x) 0.5 0.45 μ=4, σ=1 0.4 0.35 μ=4, σ=2 0.3 μ=4, σ=3 0.25 μ=4, σ=4 0.2 0.15 0.1 0.05 0 x 0510 μ and σ are the lognormal’s mean and std dev, not those of the associated normal © onntiW8 020 5 IX.