scholarly journals FORMULA OF INVERSION FOR RATIONAL CHARACTERISTIC FUNCTIONS OF PROBABILITY DISTRIBUTIONS

2018 ◽  
pp. 7-22
Author(s):  
Georgiy Aleksandrovich Popov
Keyword(s):  
Distribution Function ◽  
Probability Distributions ◽  
Rational Functions ◽  
Distribution Functions ◽  
Characteristic Functions ◽  
Arbitrary Distribution ◽  
Abscissa Axis ◽  
Entire Axis ◽  
Inversion Formulas ◽  
Gamma Distributions

The paper deals with the problem of clarifying the well-known inversion formulas for distribution functions, usually describing the increment of these functions. The validity of the corresponding inversion formulas for the distribution function π and their densities has been proved for the particular case of distributions with rational characteristic functions. The obtained formulas for distribution functions, which include additionally constant terms equal to 0.5, were not previously known. Functions of positively distributed random variables and quantities distributed over the entire axis have been considered separately. To test the hypothesis of fairness of the obtained treatment formula, including a previously unknown term equal to 0.5, in the general case there have been given examples of calculating distribution functions, whose characteristic functions are not considered as rational functions: for constant and uniform laws. The verification confirmed the objectiveness of the formulated hypothesis about the obtained validity of the inversion form for arbitrary distribution functions. It has also been shown that any distribution function and any density can be represented as a limit of a mixture of gamma distributions (distribution functions and densities), having shifts along the abscissa axis and, possibly, with altered signs of the arguments. The obtained result proves that the set of gamma distributions with shifted arguments is uniformly dense in the set of all distributions.

scholarly journals MODELING OF MULTICHANNEL QUEUING SYSTEMS AS COMPONENTS OF SOCIOTECHNICAL SYSTEMS

2018 ◽  
pp. 109-116
Author(s):  
Georgiy Aleksandrovich Popov
Keyword(s):  
Distribution Function ◽  
Recurrence Relations ◽  
Initial Distribution ◽  
Individual Characteristics ◽  
Distribution Functions ◽  
Technical System ◽  
Switching Function ◽  
Practical Implementation ◽  
Conflict Situations ◽  
Gamma Distributions

For a multichannel queuing system, in which all calls have individual characteristics of arrival and maintenance in accordance with their characteristics in the required socio-technical system and switching from one call to another are performed in accordance with a specified switching function determined by the resolution policy adopted in the socio-technical system of conflict situations, recurrence relations have been obtained for the queue lengths, for the list of free instrument changes, for the list of call numbers waiting for service, and for a number of other characteristics at successive call termination services. The procedure of sequential calculation of all specified characteristics of the system is described taking into account their internal interrelation. In accordance with this procedure, in a strictly defined sequence, eleven characteristics of the system are calculated on the basis of recurrence relations. To increase the efficiency of the process of practical implementation of the modeling procedure, it is proposed to replace the initial distribution functions of random variables by their approximate values, which are mixtures of gamma distributions whose values can be calculated significantly faster than the values of the initial distributions. The problem of finding a set of exponential distributions for a given simulated distribution function is formalized, the mixture of which approximates the distribution function with a given accuracy.

scholarly journals THE IMPACT OF SPATIAL AND TEMPORAL RESOLUTIONS IN TROPICAL SUMMER RAINFALL DISTRIBUTION: PRELIMINARY RESULTS

2017 ◽  
Vol IV-4/W2 ◽  
pp. 183-191
Author(s):  
Q. Liu ◽  
L. S Chiu ◽  
X. Hao
Keyword(s):  
Climate Models ◽  
Hydrological Cycle ◽  
Probability Distributions ◽  
Summer Rainfall ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Statistical Parameters ◽  
Spatial And Temporal Scales ◽  
Gamma Distributions ◽  
The Impact

The abundance or lack of rainfall affects peoples’ life and activities. As a major component of the global hydrological cycle (Chokngamwong & Chiu, 2007), accurate representations at various spatial and temporal scales are crucial for a lot of decision making processes. Climate models show a warmer and wetter climate due to increases of Greenhouse Gases (GHG). However, the models’ resolutions are often too coarse to be directly applicable to local scales that are useful for mitigation purposes. Hence disaggregation (downscaling) procedures are needed to transfer the coarse scale products to higher spatial and temporal resolutions. The aim of this paper is to examine the changes in the statistical parameters of rainfall at various spatial and temporal resolutions. The TRMM Multi-satellite Precipitation Analysis (TMPA) at 0.25 degree, 3 hourly grid rainfall data for a summer is aggregated to 0.5,1.0, 2.0 and 2.5 degree and at 6, 12, 24 hourly, pentad (five days) and monthly resolutions. The probability distributions (PDF) and cumulative distribution functions(CDF) of rain amount at these resolutions are computed and modeled as a mixed distribution. Parameters of the PDFs are compared using the Kolmogrov-Smironov (KS) test, both for the mixed and the marginal distribution. These distributions are shown to be distinct. The marginal distributions are fitted with Lognormal and Gamma distributions and it is found that the Gamma distributions fit much better than the Lognormal.

Certain Fourier Transforms of Distributions

1951 ◽  
Vol 3 ◽  
pp. 140-144 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szasz
Keyword(s):  
Probability Distribution ◽  
Characteristic Function ◽  
Fourier Transforms ◽  
Probability Distributions ◽  
Distribution Functions ◽  
Characteristic Functions

Fourier transforms of distribution functions are frequently studied in the theory of probability. In this connection they are called characteristic functions of probability distributions. It is often of interest to decide whether a given function φ(t) can be the characteristic function of a probability distribution, that is, whether it admits the representation

On the Decomposition of A Class of Functions of Bounded Variation

1964 ◽  
Vol 16 ◽  
pp. 479-484 ◽  
Author(s):  
R. G. Laha
Keyword(s):  
Distribution Function ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Distribution Functions ◽  
Characteristic Functions ◽  
Functions Of Bounded Variation ◽  
Image Position ◽  
Class Of Functions

Let F1(x) and F2(x) be two distribution functions, that is, non-decreasing, right-continuous functions such that Fj(— ∞) = 0 and Fj(+ ∞) = 1 (j = 1, 2). We denote their convolution by F(x) so thatthe above integrals being defined as the Lebesgue-Stieltjes integrals. Then it is easy to verify (2, p. 189) that F(x) is a distribution function. Let f1(t), f2(t), and f(t) be the corresponding characteristic functions, that is,

scholarly journals Deviations from Michaelis–Menten kinetics. Computation of the probabilities of obtaining complex curves from simple kinetic schemes

1981 ◽  
Vol 193 (1) ◽  
pp. 339-352 ◽  
Author(s):  
F Solano-Muñoz ◽  
P B McGinlay ◽  
R Woolfson ◽  
W G Bardsley
Keyword(s):  
Probability Distributions ◽  
Rational Functions ◽  
Distribution Functions ◽  
Rate Equations ◽  
British Library ◽  
Curve Shape ◽  
Product Release ◽  
Complex Curves ◽  
Probability Distribution Functions ◽  
Approximate Magnitude

1. It is possible to calculate the intrinsic probability associated with any curve shape that is allowed for rational functions of given degree when the coefficients are independent or dependent random variables with known probability distributions. 2. Computations of such probabilities are described when the coefficients of the rational function are generated according to several probability distribution functions and in particular when rate constants are varied randomly for several simple model mechanisms. 3. It is concluded that each molecular mechanism is associated with a specific set of curve-shape probabilities, and this could be of value in discriminating between model mechanisms. 4. It is shown how a computer program can be used to estimate the probability of new complexities such as extra inflexions and turning points as the degree of rate equations increases. 5. The probability of 3 : 3 rate equations giving 2 : 2 curve shapes is discussed for unrestricted coefficients and also for the substrate-modifier mechanisms. 6. The probability associated with the numerical values of coefficients in rate equations is also calculated for this mechanism, and a possible method for determining the approximate magnitude of product-release steps is given. 7. The computer programs used in the computations have been deposited as Supplement SUP 50113 (21 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem, J. (1978) 169, 5.

scholarly journals On the Calculation of the Generalised Poisson Function

1967 ◽  
Vol 4 (2) ◽  
pp. 120-128 ◽  
Author(s):  
Erkki Pesonen
Keyword(s):  
Distribution Function ◽  
Edgeworth Expansion ◽  
Distribution Functions ◽  
International Congress ◽  
Characteristic Functions ◽  
Real Numbers ◽  
Methods Of Calculation ◽  
Poisson Function ◽  
Component Representation ◽  
Special Meeting

Drs. H. Bohman and F. Esscher have reported in a recent paper) an extensive research performed in Sweden on the different methods of calculation of the distribution function of the total amount of claims. In the present paper certain methods are discussed in so far as they are different from those presented in the above quoted paper. The consideration is restricted to the generalised Poisson function even though some results can be easily extended. The author has already commented on some of the results represented in the sequel at a special meeting of the 17th International Congress of Actuaries in Edinburgh.I. Lemma. Let be the generalised Poisson function under investigation. If aiSi(x), where Σ ai = 1 (the functions Si need not be distribution functions, neither must the constants ai be real numbers of interval [0,1]), thenF(x; n, S) = F(.; a1n, S1) * …*F(.; arn, Sr) (x),as is easily verified by the use of characteristic functions. This component representation is repeatedly used in the sequel.2. A Modified Esscher Method. The Esscher method is based on an observation that the well-known Edgeworth expansion is more advantageously applicable to a conveniently modified distribution function instead of the original generalised Poisson function. Let us assume that the value of F(x) is required at a point

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

scholarly journals Verb valency in interlanguage: An extension to valency theory and new perspective on L2 learning

2020 ◽  
Vol 56 (2) ◽  
pp. 339-363
Author(s):  
Qianying Zhao ◽  
Jingyang Jiang
Keyword(s):  
Probability Distributions ◽  
Specific Property ◽  
Distribution Functions ◽  
Target Language ◽  
Native Languages ◽  
Language Competence ◽  
English Writing ◽  
Chinese Learners ◽  
L2 Learning ◽  
New Perspective

AbstractValency theory has been applied to investigate various languages, such as German, Chinese and English. However, most studies in this field were based on the linguistic materials produced by native speakers. The current research aimed to examine the valency structures in the interlanguage. Based on the English writing produced by L2 Chinese learners, we adopted the quantitative approach, trying to find out whether the distributional features of verb valency in the interlanguage also had regular probability distributions as those in the native languages, and whether there was a relationship between these valency distributional characteristics and L2 learners’ language competence. It was found that (1) verb valency in the interlanguage followed distributional regularities which had been identified in the native languages; (2) the valency features showed differences in the diversity of valency patterns, the use of valences and the complexity of forms of complements between the interlanguage and the target language; (3) the distribution functions and parameters related to verb valency could manifest the development of students’ language competence. The current research has extended valency theory to the study of interlanguage and the valency perspective has profound methodological and pedagogical implications for L2 learning. Its item-specific property and the integration of grammatical and lexical factors are conducive to analyzing the way various words combine with each other.

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