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Hoeffding's Lemma
In probability theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable. It is named after the Finnish–American mathematical statistician Wassily Hoeffding. The proof of Hoeffding's lemma uses Taylor's theorem and Jensen's inequality. Hoeffding's lemma is itself used in the proof of McDiarmid's inequality. Statement of the lemma Let ''X'' be any real-valued random variable such that a \leq X \leq b almost surely, i.e. with probability one. Then, for all \lambda \in \mathbb, :\mathbb \left e^ \right\leq \exp \Big(\lambda\mathbb \frac \Big), or equivalently, :\mathbb \left e^ \right\leq \exp \Big(\frac \Big). Proof Without loss of generality, by replacing X by X - \mathbb /math>, we can assume \mathbb = 0, so that a \leq 0 \leq b. Since e^ is a convex function of x, we have that for all x \in ,b/math>, :e^\leq \frace^+\frace^ So, :\begin \mathbb\left ^\right&\leq \frace^+\frace^\\ &= \frace^ + \frace ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment-generating Function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions. As its name implies, the moment-generating function can be used to compute a distribution’s moments: the ''n''th moment about 0 is the ''n''th derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases. The moment-generating func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Function
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. A function that is ''not'' bounded is said to be unbounded. If ''f'' is real-valued and ''f''(''x'') ≤ ''A'' for all ''x'' in ''X'', then the function is said to be bounded (from) above by ''A''. If ''f''(''x'') ≥ ''B'' for all ''x'' in ''X'', then the function is said to be bounded (from) below by ''B''. A real-valued function is bounded if and only if it is bounded from above and below. An important special case is a bounded sequence, where ''X'' is taken to be the set N of natural numbers. Thus a sequence ''f'' = (''a''0, ''a''1, ''a''2, ...) is bounded if there exists a real number ''M'' such that :, a_n, \le M for every natural number ''n''. The set of all bounded sequences forms the sequence space l^\infty. The definition of bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finnish People
Finns or Finnish people ( fi, suomalaiset, ) are a Baltic Finnic ethnic group native to Finland. Finns are traditionally divided into smaller regional groups that span several countries adjacent to Finland, both those who are native to these countries as well as those who have resettled. Some of these may be classified as separate ethnic groups, rather than subgroups of Finns. These include the Kvens and Forest Finns in Norway, the Tornedalians in Sweden, and the Ingrian Finns in Russia. Finnish, the language spoken by Finns, is closely related to other Balto-Finnic languages, e.g. Estonian and Karelian. The Finnic languages are a subgroup of the larger Uralic family of languages, which also includes Hungarian. These languages are markedly different from most other languages spoken in Europe, which belong to the Indo-European family of languages. Native Finns can also be divided according to dialect into subgroups sometimes called ''heimo'' (lit. ''tribe''), although suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territories, nine Minor Outlying Islands, and 326 Indian reservations. The United States is also in free association with three Pacific Island sovereign states: the Federated States of Micronesia, the Marshall Islands, and the Republic of Palau. It is the world's third-largest country by both land and total area. It shares land borders with Canada to its north and with Mexico to its south and has maritime borders with the Bahamas, Cuba, Russia, and other nations. With a population of over 333 million, it is the most populous country in the Americas and the third most populous in the world. The national capital of the United States is Washington, D.C. and its most populous city and principal financial center is New York City. Paleo-Americ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory. Introduction Statistical data collection is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data often follows the study protocol specified prior to the study being conducted. The data from a study can also be analyzed to consider secondary hypotheses inspired by the initial results, or to suggest new studies. A secondary analysis of the data from a planned study uses tools from data analysis, and the process of doing this is mathematical statistics. Data analysis is divided into: * descriptive statistics - the part of st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wassily Hoeffding
Wassily Hoeffding (June 12, 1914 – February 28, 1991) was a Finnish statistician and probabilist. Hoeffding was one of the founders of nonparametric statistics, in which Hoeffding contributed the idea and basic results on U-statistics. In probability theory, Hoeffding's inequality provides an upper bound on the probability for the sum of random variables to deviate from its expected value. Personal life Hoeffding was born in Mustamäki, Finland, (Gorkovskoye, Russia since 1940), although his place of birth is registered as St. Petersburg on his birth certificate. His father was an economist and a disciple of Peter Struve, the Russian social scientist and public figure. His paternal grandparents were Danish and his father's uncle was the Danish philosopher Harald Høffding. His mother, née Wedensky, had studied medicine. Both grandfathers had been engineers. In 1918 the family left Tsarskoye Selo for Ukraine and, after traveling through scenes of civil war, finally left Russ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor's Theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jensen's Inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations. Jensen's inequality generalizes the statement that the secant line of a convex function lies ''above'' the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for ''t'' ∈ ,1, :t f(x_1) + (1-t) f(x_2), while t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McDiarmid's Inequality
In probability theory and theoretical computer science, McDiarmid's inequality is a concentration inequality which bounds the deviation between the sampled value and the expected value of certain functions when they are evaluated on independent random variables. McDiarmid's inequality applies to functions that satisfy a ''bounded differences'' property, meaning that replacing a single argument to the function while leaving all other arguments unchanged cannot cause too large of a change in the value of the function. =Statement= A function f: \mathcal_1 \times \mathcal_2 \times \cdots \times \mathcal_n \rightarrow \mathbb satisfies the ''bounded differences property'' if substituting the value of the ith coordinate x_i changes the value of f by at most c_i. More formally, if there are constants c_1, c_2, \dots, c_n such that for all i\in /math>, and all x_1\in \mathcal_1,\,x_2\in \mathcal_2,\, \ldots,\, x_n \in \mathcal_n, : \sup_ \left, f(x_1, \dots, x_, x_i, x_, \ldots, x_n) - f(x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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