Sigma-algebra. If X = {a, b, c, d}, one possible σ-algebra on X is Σ = {∅, {a, b}, {c, d}, {a, b, c, d}}, where ∅ is the empty set. However, a finite algebra is always a σ-algebra. If {A1, A2, A3, …} is a countable partition of X then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra. A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy). Motivation[edit] There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets. Measure[edit] One would like to assign a size to every subset of X, but in many natural settings, this is not possible.
Observe that then or. Lévy process. The most well known examples of Lévy processes are Brownian motion and the Poisson process. Aside from Brownian motion with drift, all other Lévy processes, except the deterministic case, have discontinuous paths. Mathematical definition[edit] A stochastic process is said to be a Lévy process if it satisfies the following properties: If is a Lévy process then one may construct a version of such that is almost surely right continuous with left limits.
Properties[edit] Independent increments[edit] Stationary increments[edit] To call the increments stationary means that the probability distribution of any increment Xt − Xs depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed. is a Wiener process, the probability distribution of Xt − Xs is normal with expected value 0 and variance t − s.
Infinite divisibility[edit] , there is a Lévy process such that the law of is given by Moments[edit] Lévy–Khintchin representation[edit] where . .
Rencontres-Zahl. In der Kombinatorik versteht man unter einer Rencontres-Zahl ( französisch Begegnungen ) die mit bezeichnete Anzahl der Permutationen einer Menge unterscheidbarer Elemente, bei der genau Elemente ihren ursprünglichen Platz beibehalten bzw. rein zufällig „wiederfinden“: Für den Fall, dass keines der Elemente seinen Platz beibehält bzw. „wiederfindet“, ergibt sich als Sonderfall die Subfakultät , eine Formel für die Zahl möglicher fixpunktfreier Permutationen (auch Derangements oder „Totalversetzungen“) der Elemente, bei denen also keines von ihnen an seinem bisherigen Platz bleibt: Beispiel [ Bearbeiten ] Ein Autobesitzer hat den Motor seines neuen Vierzylinders geputzt und vergessen, sich zu notieren, welches der vier Zündkabel auf welche Zündkerze gehört.
Im Detail: Ein Jahr später passiert ihm dasselbe mit dem Motor seines neuen Sechszylinders. Literatur [ Bearbeiten ] Subfakultät. Die Subfakultät ist eine vornehmlich in der Kombinatorik auftretende Funktion . Sie gibt die Anzahl der fixpunktfreien Permutationen einer Menge mit Elementen an und wird durch notiert. Die Subfakultät ist eng mit der Fakultät verwandt, die die Gesamtzahl der Permutationen einer -elementigen Menge angibt.
Sie ist näherungsweise gleich dem Quotient aus der Fakultät und der eulerschen Zahl Definition [ Bearbeiten ] Die Subfakultät einer natürlichen Zahl wird mit Hilfe der Fakultät durch definiert. Entspricht der Anzahl der fixpunktfreien Permutationen (Derangements) einer -elementigen Menge, während die Fakultät die Anzahl aller möglichen Permutationen angibt. Beispiel [ Bearbeiten ] Angenommen, man hat sechs verschiedenfarbige Kugeln, und zu jeder Kugel ein Kästchen in der passenden Farbe. Möglichkeiten. Weitere Darstellungen [ Bearbeiten ] Rundungsdarstellungen [ Bearbeiten ] Es gilt mit der eulerschen Zahl und der unvollständigen Gammafunktion . Gerundet erhält man für sogar die exakte Formel wobei die und.
Fixpunktfreie Permutation. Graph einer fixpunktfreien Permutation der Zahlen von 1 bis 8. Durch die Permutation wird keine der Zahlen festgehalten. Eine fixpunktfreie Permutation oder Derangement (von französisch déranger „durcheinanderbringen“) ist in der Kombinatorik eine Permutation der Elemente einer Menge, sodass kein Element seine Ausgangsposition beibehält. Die Anzahl möglicher fixpunktfreier Permutationen einer Menge mit Elementen wird durch die Subfakultät angegeben. Strebt der Anteil der fixpunktfreien Permutationen sehr schnell gegen den Kehrwert der eulerschen Zahl . Ausgangsproblem [ Bearbeiten ] Der französische Mathematiker Pierre Rémond de Montmort stellte Anfang des 18. Ein Spieler mischt einen Satz von 13 Spielkarten einer Farbe und legt ihn als Stapel vor sich hin. Nun stellt de Montmort sich die Frage nach der Wahrscheinlichkeit , mit der der Spieler das Spiel gewinnt.
Liegt. Zwei Spieler besitzen jeweils ein vollständiges Kartenspiel mit 52 Karten. Definition [ Bearbeiten ] Ist fixpunktfrei, wenn . Prinzip von Inklusion und Exklusion. Das Prinzip von Inklusion und Exklusion (auch Prinzip der Einschließung und Ausschließung oder Einschluss-/Ausschluss-Verfahren) ist eine hilfreiche Technik zur Bestimmung der Mächtigkeit einer Menge . Sie findet vor allem in der Kombinatorik und in der Zahlentheorie Anwendung. Das Prinzip drückt dazu die Kardinalität einer Ursprungsmenge durch die Kardinalitäten ihrer Teilmengen aus. Diese sind in aller Regel einfacher zu bestimmen. Von oben abgeschätzt wird ( Inklusion ), anschließend jedoch durch die Subtraktion der Größe des gemeinsamen Schnittes der Teilmengen dies wieder zu korrigieren versucht wird ( Exklusion ). Das Prinzip [ Bearbeiten ] Prinzip von Inklusion und Exklusion am Beispiel von drei Mengen Es ist ein bekanntes Ergebnis, dass für je zwei endliche Mengen und gilt. Wird zunächst die Kardinalität von von oben abgeschätzt.
Korrigiert. Als auch in enthalten sind und somit zunächst doppelt gezählt wurden. Im Allgemeinen wollen wir die Kardinalität der Vereinigung von endlichen Mengen. Set (mathematics) An example of a Venn diagram The intersection of two sets is made up with the objects contained in both sets In mathematics , a set is a collection of distinct objects, considered as an object in its own right . For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}.
Sets are one of the most fundamental concepts in mathematics . Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education , elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
Definition [ edit ] A set is a well defined collection of objects. Sets are conventionally denoted with capital letters . Describing sets [ edit ] B is the set of colors of the French flag . Example: Inclusion–exclusion principle. The union of sets A and B This formula can be verified by counting how many times each region in the Venn diagram figure is included in the right-hand side of the formula. In this case, when removing the contributions of over-counted elements, the number of elements in the mutual intersection of the three sets has been subtracted too often, so must be added back in to get the correct total.
Inclusion–exclusion illustrated by a Venn diagram for three sets As finite probabilities are computed as counts relative to the cardinality of the probability space, the formulas for the principle of inclusion–exclusion remain valid when the cardinalities of the sets are replaced by finite probabilites. More generally, both versions of the principle can be put under the common umbrella of measure theory. "One of the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of inclusion–exclusion. Statement[edit] This can be compactly written as . . Zipf's law. Zipf's law /ˈzɪf/, an empirical law formulated using mathematical statistics, refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions.
The law is named after the American linguist George Kingsley Zipf (1902–1950), who first proposed it (Zipf 1935, 1949), though the French stenographer Jean-Baptiste Estoup (1868–1950) appears to have noticed the regularity before Zipf.[1] It was also noted in 1913 by German physicist Felix Auerbach[2] (1856–1933). Motivation[edit] Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table. Thus the most frequent word will occur approximately twice as often as the second most frequent word, three times as often as the third most frequent word, etc. Theoretical review[edit] Formally, let: Related laws[edit] Zipf–Mandelbrot law. In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoît Mandelbrot, who subsequently generalized it.
The probability mass function is given by: where is given by: which may be thought of as a generalization of a harmonic number. In the formula, k is the rank of the data, and q and s are parameters of the distribution. In the limit as approaches infinity, this becomes the Hurwitz zeta function . And the Zipf–Mandelbrot law becomes Zipf's law. It becomes a Zeta distribution.
Applications[edit] The distribution of words ranked by their frequency in a random text corpus is generally a power-law distribution, known as Zipf's law. Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandlebrot distributions.[2] Notes[edit] Distance correlation. These distance-based measures can be put into an indirect relationship to the ordinary moments by an alternative formulation (described below) using ideas related to Brownian motion, and this has led to the use of names such as Brownian covariance and Brownian distance covariance.
Several sets of (x, y) points, with the Distance correlation coefficient of x and y for each set. Compare to the graph on correlation Background[edit] The classical measure of dependence, the Pearson correlation coefficient,[1] is mainly sensitive to a linear relationship between two variables. Distance correlation was introduced in 2005 by Gabor J Szekely in several lectures to address this deficiency of Pearson’s correlation, namely that it can easily be zero for dependent variables. Correlation = 0 (uncorrelatedness) does not imply independence while distance correlation = 0 does imply independence. Definitions[edit] Distance covariance[edit] Let us start with the definition of the sample distance covariance. Prokhorov's theorem. In measure theory Prokhorov’s theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces.
The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements. Statement of the theorem[edit] Let be a separable metric space. Let denote the collection of all probability measures defined on (with its Borel σ-algebra). Theorem. Corollaries[edit] For Euclidean spaces we have that: If is a tight sequence in (the collection of probability measures on -dimensional Euclidean space), then there exist a subsequence and a probability measure such that converges weakly to .If is a tight sequence in such that every weakly convergent subsequence has the same limit , then the sequence converges weakly to . Extension[edit] Theorem: Suppose that.
Lévy–Prokhorov metric. Definition[edit] Let be a metric space with its Borel sigma algebra . Let denote the collection of all probability measures on the measurable space For a subset , define the ε-neighborhood of by where is the open ball of radius centered at The Lévy–Prokhorov metric is defined by setting the distance between two probability measures and to be For probability measures clearly Some authors omit one of the two inequalities or choose only open or closed ; either inequality implies the other, and , but restricting to open sets may change the metric so defined (if is not Polish). Properties[edit] See also[edit] References[edit] Martingale (probability theory) Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. To contrast, in a process that is not a martingale, it may still be the case that the expected value of the process at one time is equal to the expected value of the process at the next time.
However, knowledge of the prior outcomes (e.g., all prior cards drawn from a card deck) may be able to reduce the uncertainty of future outcomes. Thus, the expected value of the next outcome given knowledge of the present and all prior outcomes may be higher than the current outcome if a winning strategy is used.
Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games. A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for any time n, or which states that the average "winnings" from observation are 0. Fat-tailed distribution. A fat-tailed distribution is a probability distribution that has the property, along with the other heavy-tailed distributions, that it exhibits large skewness or kurtosis.
This comparison is often made relative to the normal distribution, or to the exponential distribution. Fat-tailed distributions have been empirically encountered in a variety of areas: economics, physics, and earth sciences. Some fat-tailed distributions have power law decay in the tail of the distribution, but do not necessarily follow a power law everywhere.[1] Definition[edit] A variety of Cauchy distributions for various location and scale parameters.
The distribution of a random variable X is said to have a fat tail if That is, if X has probability density function Here the tilde notation " " refers to the asymptotic equivalence of functions. Fat tails and risk estimate distortions[edit] Levy flight from a Cauchy Distribution compare to the Brownian Motion (below). Applications in economics[edit] See also[edit] Kolmogorov equations. Kolmogorov's zero–one law. Kolmogorov's inequality. Kolmogorov–Smirnov test. Kolmogorov's criterion. Keno. Ewens's sampling formula. Watterson estimator. Coupon collector's problem. Probability space. Borel–Kolmogorov paradox. Chapman–Kolmogorov equation. Anderson–Darling test. Shapiro–Wilk test. Brownian motion. Master equation.
Fokker–Planck equation. Ising-Modell. Kolmogorov equations (Markov jump process) Stationary ergodic process. Wahrscheinlichkeitstheorie. P-value. Probability axioms.