Distributions
In mathematics, distributions refer to a generalization of functions that can act on test functions to produce real numbers. They are used to extend the concept of derivatives to a wider class of functions, including those that are not necessarily smooth or continuous. Distributions are fundamental in the study of partial differential equations and functional analysis.
4 Key excerpts on "Distributions"
eBook - ePub- Umberto Cherubini, Giovanni Della Lunga, Sabrina Mulinacci, Pietro Rossi(Authors)
- 2010(Publication Date)
- Wiley(Publisher)
...It is a continuous analogue of the discrete Kronecker delta. In the context of signal processing it is often referred to as the unit impulse function. In finance, we saw in Chapter 4 that it is the limit of a sequence of butterfly spreads with the same payoff but strike prices closer and closer. Notice that the Dirac delta is not strictly a function, while for many purposes it can be manipulated as such; formally it can be correctly defined as a distribution as follows: A helpful identity is the scaling property (taking α non-zero), where in the third step we have put u = lαlx, so: The scaling property may be generalized to: where x i are the real roots of g(x) (assumed simple roots) and, Thus, for example, In the integral form the generalized scaling property may be written as In an n -dimensional space with position vector r, this is generalized to: where the integral on the right is over ∂V, the n —1 dimensional surface defined by g(r) = 0. The integral of the time-delayed Dirac delta is given by: (the shifting property). The delta function is said to “shift out” the value at t = T. 5.4 THE CALCULUS OF Distributions The power of distributional analysis in large part rests on the facts that every distribution possesses derivatives of all orders and that differentiation is a continuous operation in this theory. As a consequence, distributional differentiation commutes with various limiting processes such as infinite summation and integration. This is in contrast to classical analysis wherein either such operations cannot be interchanged or the inversion of order must be justified by additional arguments. 5.4.1 Distribution derivative To define the derivative of a distribution, we first consider the case of a differentiable and integrable function f : R → R. If ϕ is a test function, then we have using integration by parts (note that ϕ is zero outside of a bounded set and that therefore no boundary values have to be taken into account)...
eBook - ePubInterpreting Statistics for Beginners
A Guide for Behavioural and Social Scientists
- Vladimir Hedrih, Andjelka Hedrih(Authors)
- 2022(Publication Date)
- Routledge(Publisher)
...4 Distributions DOI: 10.4324/9781003107712-4 4.1 Theoretical and empirical Distributions As stated in the previous chapter, a distribution is a listing or a function describing all the existing (or all possible) values of a variable and how often they occur. It is usually created by listing each specific value of the variable and the frequency with which it occurs, but it can alternatively be also presented by creating intervals of values for continuous variables or broader categories for variables with many different possible values (such as continuous variables) and then listing the frequencies of entities in each of these intervals or broader categories. It can also be presented as a probability density function. Aside from providing a simple factual description of values of sample entities on considered variables, Distributions are also used in statistics to make various inferences about the circumstances around properties of the variable, the sample and the measurement based on the shape of the distribution. This is done by comparing the distribution obtained from the empirical data to certain known shapes of Distributions, Distributions that are known to be the result of certain known processes. If the shape of the sample distribution resembles the shape it is compared with, we can infer, provided other properties of the sample and the measurement allow it, that processes known to result in such Distributions likely took place. Or otherwise, we can conclude that our distribution likely was not the results of processes that result in Distributions of a certain shape. For this reason, we will make a distinction between empirical and theoretical Distributions: An empirical distribution is a real distribution we have obtained from real data i.e. from real observations or measurements...
- Michael B. Miller(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...CHAPTER 4 Distributions I n Chapter 2, we were introduced to random variables. In nature and in finance, random variables tend to follow certain patterns, or Distributions. In this chapter we will learn about some of the most widely used probability Distributions in risk management. PARAMETRIC Distributions Distributions can be divided into two broad categories: parametric Distributions and nonparametric Distributions. A parametric distribution can be described by a mathematical function. In the following sections we explore a number of parametric Distributions, including the uniform distribution and the normal distribution. A nonparametric distribution cannot be summarized by a mathematical formula. In its simplest form, a nonparametric distribution is just a collection of data. An example of a nonparametric distribution would be a collection of historical returns for a security. Parametric Distributions are often easier to work with, but they force us to make assumptions, which may not be supported by real-world data. Nonparametric Distributions can fit the observed data perfectly. The drawback of nonparametric Distributions is that they are potentially too specific, which can make it difficult to draw any general conclusions. UNIFORM DISTRIBUTION For a continuous random variable, X, recall that the probability of an outcome occurring between b 1 and b 2 can be found by integrating as follows: where f (x) is the probability density function (PDF) of X. The uniform distribution is one of the most fundamental Distributions in statistics...
eBook - ePubStatistics in Psychology
An Historical Perspective
- Michael Cowles(Author)
- 2005(Publication Date)
- Psychology Press(Publisher)
...6 Distributions When Graunt and Halley and Quetelet made their inferences, they made them on the basis of their examination of frequency Distributions. Tables, charts, and graphs – no matter how the information is displayed – all can be used to show a listing of data, or classifications of data, and their associated frequencies. These are frequency Distributions. By extension, such depictions of the frequency of occurrence of observations can be used to assess the expectation of particular values, or classes of values, occurring in the future. Real frequency Distributions can then be used as probability Distributions. In general, however, the probability Distributions that are familiar to the users of statistical techniques are theoretical Distributions, abstractions based on a mathematical rule, that match, or approximate, Distributions of events in the real world. When bodies of data are described, it is the graph and the chart that are used. But the theoretical Distributions of statistics and probability theory are described by the mathematical rules or functions that define the relationships between data, both real and hypothetical, and their expected frequencies or probabilities. Over the last 300 years or so, the characteristics of a great many theoretical Distributions, all of which have been found to have some practical utility in one situation or another, have been examined. The following discussion is limited to three Distributions that are familiar to users of basic statistics in psychology. An account of some fundamental sampling Distributions is given later. The Binomial Distribution In the years 1665–1666, when Isaac Newton was 23 and had just earned his degree, his Cambridge college (Trinity) was closed because of the plague. Newton went home to Woolsthorpe in Lincolnshire and began, in peace and leisure, a scientific revolution...
