DOCSLIB.ORG

Fundamental Properties of Generalized Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fundamental Properties of Generalized Functions Appendix Fundamental Properties of Generalized Functions A.1 Definition of generalized functions First of all, let us give some material from mathematics, which is necessary for defining the concept of a generalized function. Let us consider the following linear functional: T [ϕ(x)] = f (x)ϕ(x)dx. (A.1) Here ϕ(x) is the argument of the functional, and f (x) is its kernel, which deter- mines the properties of the functional. Almost everywhere below, we consider these functions being real, mapping points x of the number line R onto R. Thereby, the linear functional T [ϕ(x)] maps the “points” ϕ(x) of a certain functional space D (ϕ(x) ∈ D) onto points of the real line R. Let us call the above mentioned “func- tional points” ϕ(x) test functions, and D —aspace of test functions. Let us also note that indefinite integrals, in our notation, symbolize definite integrals over the entire x-axis, i.e. having the integration limits ±∞. As soon as the space D of test functions is chosen, the conjugate to it set D of linear continuous functionals T [ϕ], underlying the following definition of general- ized functions, becomes automatically defined. The space D of test functions is subject to several natural requirements: it needs to be sufficiently large, in order to identify any continuous kernel f (x) through val- ues of the integral (A.1). In other words, as soon as values of the functional T [ϕ] are known for a certain set of test functions ϕ ∈ D, values of the kernel f (x) may be defined for all x. On the other hand, the space D of test functions is subjected to quite strict limitations, in order to obtain as rich a set D of linear continuous functions T [ϕ] as possible. It so happens that the family of infinitely differentiable compact functions is a suitable candidate for the space D. Below, we reserve the notation D precisely for this functional space. Recall that a function ϕ(x) is called compact, if it has a bounded support. In its turn, the support supp ϕ of a function ϕ(x) is the closure of the set of the points on the x-axis, where ϕ(x) = 0. Now we are fully prepared to formulate the definition of a generalized function. 442 Fundamental Properties of Generalized Functions By definition: any linear functional T [ϕ], which is continuous on the set D of infinitely differentiable compact functions is called a generalized function. Let us give a few comments on the above-formulated definition of a generalized function: 1. Recall that a functional T [ϕ] is called linear, if, for all test functions ϕ(x) ∈ D and ψ(x) ∈ D, the following equality holds: T [αϕ + βψ]=αT [ϕ]+βT[ϕ] , where α and β are arbitrary constants. 2. A functional T on D is called continuous, if for any functional sequence {ϕk(x)}, whose elements belong to the space D and converge at k → ∞ to a certain test function ϕ(x) ∈ D, the corresponding numerical sequence {T[ϕ k]} converges to the number T [ϕ]. Thereby convergence of the sequence {ϕ k(x)} is understood in the following sense: a) The supports of all elements of the sequence {ϕk} belong to some bounded set of the x-axis. → ∞ ϕ ( ) ϕ (n)( ) b) At k , the functions k x and all their derivatives k x uniformly con- verge to the function ϕ(x) and its derivatives, respectively. If it is possible to express a linear continuous functional T[ϕ(x)] in the form of the integral (A.1), whose kernel f (x) is an everywhere continuous, bounded function, then such a functional is called a regular generalized function and it is identified with the kernel f (x). In this sense, all ordinary continuous functions are regular generalized functions. There are, however, linear continuous functionals T [ϕ], which cannot be iden- tified with a certain continuous kernel f (x). In this case, such a linear continuous functional T [ϕ(x)] is called a singular generalized function. Perhaps the most important example of a singular generalized function is the functional δ[ϕ(x)] = ϕ(0), (A.2) mapping the test function ϕ(x) ∈ D onto its value at x = 0. This, obviously linear and continuous, functional is called a delta function. Although this functional cannot be expressed by means of the usual integral (A.1), it is often written in the following symbolic integral form: δ[ϕ(x)] = δ(x)ϕ(x)dx = ϕ(0), (A.3) where δ(x) is the symbol for the delta function, which is defined by its sifting prop- erty, according to which the value ϕ(0) corresponds to the “integral” in (A.3). The advantage of such a symbolic notation consists in its clarity during actual computations using delta functions. Let us illustrate the “transparency” of the sym- bolic integral notation of generalized functions, related to the delta function, by A.2 Fundamental sequences 443 expressing the functional δa[ϕ(x)] = ϕ(a) by means of the equality δ(x − a)ϕ(x)dx = ϕ(a) (A.4) via a delta function δ(x − a) with a shifted argument. In the following, we will pre- dominantly use the mathematically not quite correct, but convenient while solving applied problems, symbolic integral notation of the form (A.3), (A.4). While discussing properties of generalized functions, an important role is played by the concept of the support of a generalized function. Namely, a linear continuous functional T is considered to be equal to zero in an open domain B of the x-axis, if T [ϕ(x)] = 0 for all test functions ϕ(x), whose supports belong to the set B. The closure of the largest open domain, where a generalized function T is not equal to zero is called the support of the generalized function T , and it is denoted by supp T . From the definition, it immediately follows that the support of the delta function consists of the only point x = 0, i.e. supp δ(x)={0}. (A.5) Note that if a generalized function T is compact, i.e. it has a bounded support, then it is possible to relax the conditions, imposed on the set of test functions ϕ(x),by requiring only their infinite differentiability (ϕ ∈ C ∞). For instance, it is possible to assign in (A.6) ϕ(x) ≡ 1, which leads to the normalization condition of the delta function δ(x)dx = 1. (A.6) In conclusion, let us discuss an important for physical applications, but ignored by mathematicians, property of the delta function. It belongs to a comparatively narrow class of scale-invariant functions, whose argument may be dimensional; e.g., it may be a spatial coordinate x or time t; and which has a non-zero dimension depending on the dimension of the argument. Namely, the delta function, which has the spatial coordinate x as its argument, has the dimension of the reciprocal of x: − [δ(x)] = L 1, as it is seen from the normalization condition (A.6). Thus δ(x) has the same dimen- sion as the power function 1/x. A.2 Fundamental sequences Although the delta function is, in principle, non-representable in the integral form (A.3), it can, nevertheless, be obtained as the limit of a sequence of ordinary inte- grals Tk[ϕ(x)] = fk(x)ϕ(x)dx, (A.7) 444 Fundamental Properties of Generalized Functions corresponding to the sequence of locally integrable functions { f k(x)}. This limit is understood in the sense that, for some sequence { f k(x)}, k = 1,2,..., and for any test function ϕ(x) ∈ D, the following limiting equality holds: Tk[ϕ(x)] → ϕ(0)(k → ∞). It this is indeed so, it is said that the sequence { fk(x)} weakly converges to the delta function. The choice of a weakly converging to the delta function δ(x) sequence of regular linear functionals {Tk}, defined by their kernels { fk(x)}, is not unique. Therefore it is useful to discuss several such sequences, which give an opportunity better to understand the “internal structure” of delta functions. Example 1. Let us consider a family of Gaussian functions (see Fig. A.1) 1 x2 γε (x)=√ exp − , (A.8) 2πε 2ε depending on the parameter ε > 0, and take the sequence of functions f k(x)= γ1/k(x), k = 1,2,... as a weakly convergent sequence. Note that the factor in front of the exponential function on the right-hand side of Eq. (A.8) is chosen so that the normalization condition holds: γε (x)dx = 1. At k → ∞, the value of the parameter ε tends to zero, and the Gaussian func- tion, “approximating the delta function,” becomes progressively higher and nar- rower, “concentrating” at the origin, while conserving the total area under the Gaus- sian curve so that, for all elements of the Gaussian-function sequence, the above- mentioned normalization condition is satisfied. Example 2. Another, important for theory and applications, weakly convergent to the delta function sequence is generated by the Cauchy distribution Fig. A.1 Three elements of the sequence of Gaus- sian functions (A.8), weakly convergent to the delta function, for k = 1;2;4 (ε = 1;1/2;1/4). A.2 Fundamental sequences 445 1 ε cε (x)= . (A.9) π x2 + ε2 Although, at first sight, the profiles of the Cauchy distribution seem similar to the Gaussian curves discussed above, there are significant distinctions between them. The main one is the fact that the Cauchy distribution tends to zero at x →±∞ rather slowly, according to the power law ε cε (x) ∼ .
Load more

Recommended publications
  • The Fundamental Theorem of Calculus for Lebesgue Integral
    Divulgaciones Matem´aticasVol. 8 No. 1 (2000), pp. 75{85 The Fundamental Theorem of Calculus for Lebesgue Integral El Teorema Fundamental del C´alculo para la Integral de Lebesgue Di´omedesB´arcenas([email protected]) Departamento de Matem´aticas.Facultad de Ciencias. Universidad de los Andes. M´erida.Venezuela. Abstract In this paper we prove the Theorem announced in the title with- out using Vitali's Covering Lemma and have as a consequence of this approach the equivalence of this theorem with that which states that absolutely continuous functions with zero derivative almost everywhere are constant. We also prove that the decomposition of a bounded vari- ation function is unique up to a constant. Key words and phrases: Radon-Nikodym Theorem, Fundamental Theorem of Calculus, Vitali's covering Lemma. Resumen En este art´ıculose demuestra el Teorema Fundamental del C´alculo para la integral de Lebesgue sin usar el Lema del cubrimiento de Vi- tali, obteni´endosecomo consecuencia que dicho teorema es equivalente al que afirma que toda funci´onabsolutamente continua con derivada igual a cero en casi todo punto es constante. Tambi´ense prueba que la descomposici´onde una funci´onde variaci´onacotada es ´unicaa menos de una constante. Palabras y frases clave: Teorema de Radon-Nikodym, Teorema Fun- damental del C´alculo,Lema del cubrimiento de Vitali. Received: 1999/08/18. Revised: 2000/02/24. Accepted: 2000/03/01. MSC (1991): 26A24, 28A15. Supported by C.D.C.H.T-U.L.A under project C-840-97. 76 Di´omedesB´arcenas 1 Introduction The Fundamental Theorem of Calculus for Lebesgue Integral states that: A function f :[a; b] R is absolutely continuous if and only if it is ! 1 differentiable almost everywhere, its derivative f 0 L [a; b] and, for each t [a; b], 2 2 t f(t) = f(a) + f 0(s)ds: Za This theorem is extremely important in Lebesgue integration Theory and several ways of proving it are found in classical Real Analysis.
    [Show full text]
  • [Math.FA] 3 Dec 1999 Rnfrneter for Theory Transference Introduction 1 Sas Ihnrah U Twl Etetdi Eaaepaper
    Transference in Spaces of Measures Nakhl´eH. Asmar,∗ Stephen J. Montgomery–Smith,† and Sadahiro Saeki‡ 1 Introduction Transference theory for Lp spaces is a powerful tool with many fruitful applications to sin- gular integrals, ergodic theory, and spectral theory of operators [4, 5]. These methods afford a unified approach to many problems in diverse areas, which before were proved by a variety of methods. The purpose of this paper is to bring about a similar approach to spaces of measures. Our main transference result is motivated by the extensions of the classical F.&M. Riesz Theorem due to Bochner [3], Helson-Lowdenslager [10, 11], de Leeuw-Glicksberg [6], Forelli [9], and others. It might seem that these extensions should all be obtainable via transference methods, and indeed, as we will show, these are exemplary illustrations of the scope of our main result. It is not straightforward to extend the classical transference methods of Calder´on, Coif- man and Weiss to spaces of measures. First, their methods make use of averaging techniques and the amenability of the group of representations. The averaging techniques simply do not work with measures, and do not preserve analyticity. Secondly, and most importantly, their techniques require that the representation is strongly continuous. For spaces of mea- sures, this last requirement would be prohibitive, even for the simplest representations such as translations. Instead, we will introduce a much weaker requirement, which we will call ‘sup path attaining’. By working with sup path attaining representations, we are able to prove a new transference principle with interesting applications. For example, we will show how to derive with ease generalizations of Bochner’s theorem and Forelli’s main result.
    [Show full text]
  • 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]
  • Introduction Into Quaternions for Spacecraft Attitude Representation
    Introduction into quaternions for spacecraft attitude representation Dipl. -Ing. Karsten Groÿekatthöfer, Dr. -Ing. Zizung Yoon Technical University of Berlin Department of Astronautics and Aeronautics Berlin, Germany May 31, 2012 Abstract The purpose of this paper is to provide a straight-forward and practical introduction to quaternion operation and calculation for rigid-body attitude representation. Therefore the basic quaternion denition as well as transformation rules and conversion rules to or from other attitude representation parameters are summarized. The quaternion computation rules are supported by practical examples to make each step comprehensible. 1 Introduction Quaternions are widely used as attitude represenation parameter of rigid bodies such as space- crafts. This is due to the fact that quaternion inherently come along with some advantages such as no singularity and computationally less intense compared to other attitude parameters such as Euler angles or a direction cosine matrix. Mainly, quaternions are used to • Parameterize a spacecraft's attitude with respect to reference coordinate system, • Propagate the attitude from one moment to the next by integrating the spacecraft equa- tions of motion, • Perform a coordinate transformation: e.g. calculate a vector in body xed frame from a (by measurement) known vector in inertial frame. However, dierent references use several notations and rules to represent and handle attitude in terms of quaternions, which might be confusing for newcomers [5], [4]. Therefore this article gives a straight-forward and clearly notated introduction into the subject of quaternions for attitude representation. The attitude of a spacecraft is its rotational orientation in space relative to a dened reference coordinate system.
    [Show full text]
  • 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.
    [Show full text]
  • GENERALIZED FUNCTIONS LECTURES Contents 1. the Space
    GENERALIZED FUNCTIONS LECTURES Contents 1. The space of generalized functions on Rn 2 1.1. Motivation 2 1.2. Basic definitions 3 1.3. Derivatives of generalized functions 5 1.4. The support of generalized functions 6 1.5. Products and convolutions of generalized functions 7 1.6. Generalized functions on Rn 8 1.7. Generalized functions and differential operators 9 1.8. Regularization of generalized functions 9 n 2. Topological properties of Cc∞(R ) 11 2.1. Normed spaces 11 2.2. Topological vector spaces 12 2.3. Defining completeness 14 2.4. Fréchet spaces 15 2.5. Sequence spaces 17 2.6. Direct limits of Fréchet spaces 18 2.7. Topologies on the space of distributions 19 n 3. Geometric properties of C−∞(R ) 21 3.1. Sheaf of distributions 21 3.2. Filtration on spaces of distributions 23 3.3. Functions and distributions on a Cartesian product 24 4. p-adic numbers and `-spaces 26 4.1. Defining p-adic numbers 26 4.2. Misc. -not sure what to do with them (add to an appendix about p-adic numbers?) 29 4.3. p-adic expansions 31 4.4. Inverse limits 32 4.5. Haar measure and local fields 32 4.6. Some basic properties of `-spaces. 33 4.7. Distributions on `-spaces 35 4.8. Distributions supported on a subspace 35 5. Vector valued distributions 37 Date: October 7, 2018. 1 GENERALIZED FUNCTIONS LECTURES 2 5.1. Smooth measures 37 5.2. Generalized functions versus distributions 38 5.3. Some linear algebra 39 5.4. Generalized functions supported on a subspace 41 6.
    [Show full text]
  • Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
    Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence.
    [Show full text]
  • Rational Approximation of the Absolute Value Function from Measurements: a Numerical Study of Recent Methods
    Rational approximation of the absolute value function from measurements: a numerical study of recent methods I.V. Gosea∗ and A.C. Antoulasy May 7, 2020 Abstract In this work, we propose an extensive numerical study on approximating the absolute value function. The methods presented in this paper compute approximants in the form of rational functions and have been proposed relatively recently, e.g., the Loewner framework and the AAA algorithm. Moreover, these methods are based on data, i.e., measurements of the original function (hence data-driven nature). We compare numerical results for these two methods with those of a recently-proposed best rational approximation method (the minimax algorithm) and with various classical bounds from the previous century. Finally, we propose an iterative extension of the Loewner framework that can be used to increase the approximation quality of the rational approximant. 1 Introduction Approximation theory is a well-established field of mathematics that splits in a wide variety of subfields and has a multitude of applications in many areas of applied sciences. Some examples of the latter include computational fluid dynamics, solution and control of partial differential equations, data compression, electronic structure calculation, systems and control theory (model order reduction reduction of dynamical systems). In this work we are concerned with rational approximation, i.e, approximation of given functions by means of rational functions. More precisely, the original to be approximated function is the absolute value function (known also as the modulus function). Moreover, the methods under consideration do not necessarily require access to the exact closed-form of the original function, but only to evaluations of it on a particular domain.
    [Show full text]
  • Stability in the Almost Everywhere Sense: a Linear Transfer Operator Approach ∗ R
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 368 (2010) 144–156 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Stability in the almost everywhere sense: A linear transfer operator approach ∗ R. Rajaram a, ,U.Vaidyab,M.Fardadc, B. Ganapathysubramanian d a Dept. of Math. Sci., 3300, Lake Rd West, Kent State University, Ashtabula, OH 44004, United States b Dept. of Elec. and Comp. Engineering, Iowa State University, Ames, IA 50011, United States c Dept. of Elec. Engineering and Comp. Sci., Syracuse University, Syracuse, NY 13244, United States d Dept. of Mechanical Engineering, Iowa State University, Ames, IA 50011, United States article info abstract Article history: The problem of almost everywhere stability of a nonlinear autonomous ordinary differential Received 20 April 2009 equation is studied using a linear transfer operator framework. The infinitesimal generator Available online 20 February 2010 of a linear transfer operator (Perron–Frobenius) is used to provide stability conditions of Submitted by H. Zwart an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative Keywords: Almost everywhere stability solution for a steady state advection type linear partial differential equation. We refer Advection equation to this non-negative solution, verifying almost everywhere global stability, as Lyapunov Density function density. A numerical method using finite element techniques is used for the computation of Lyapunov density. © 2010 Elsevier Inc. All rights reserved. 1. Introduction Stability analysis of an ordinary differential equation is one of the most fundamental problems in the theory of dynami- cal systems.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • 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 ) .
    [Show full text]
  • How to Enter Answers in Webwork
    Introduction to WeBWorK 1 How to Enter Answers in WeBWorK Addition + a+b gives ab Subtraction - a-b gives ab Multiplication * a*b gives ab Multiplication may also be indicated by a space or juxtaposition, such as 2x, 2 x, 2*x, or 2(x+y). Division / a a/b gives b Exponents ^ or ** a^b gives ab as does a**b Parentheses, brackets, etc (...), [...], {...} Syntax for entering expressions Be careful entering expressions just as you would be careful entering expressions in a calculator. Sometimes using the * symbol to indicate multiplication makes things easier to read. For example (1+2)*(3+4) and (1+2)(3+4) are both valid. So are 3*4 and 3 4 (3 space 4, not 34) but using an explicit multiplication symbol makes things clearer. Use parentheses (), brackets [], and curly braces {} to make your meaning clear. Do not enter 2/4+5 (which is 5 ½ ) when you really want 2/(4+5) (which is 2/9). Do not enter 2/3*4 (which is 8/3) when you really want 2/(3*4) (which is 2/12). Entering big quotients with square brackets, e.g. [1+2+3+4]/[5+6+7+8], is a good practice. Be careful when entering functions. It is always good practice to use parentheses when entering functions. Write sin(t) instead of sint or sin t. WeBWorK has been programmed to accept sin t or even sint to mean sin(t). But sin 2t is really sin(2)t, i.e. (sin(2))*t. Be careful. Be careful entering powers of trigonometric, and other, functions.
    [Show full text]