Martingale representation theoremIn probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion. The theorem only asserts the existence of the representation and does not help to find it explicitly; it is possible in many cases to determine the form of the representation using Malliavin calculus. Similar theorems also exist for martingales on filtrations induced by jump processes, for example, by Markov chains.
Processus de branchementEn théorie des probabilités, un processus de branchement est un processus stochastique formé par une collection de variables aléatoires. Les variables aléatoires d'un processus stochastique sont indexées par les nombres entiers naturels. Les processus de branchement ont été développés en premier lieu pour décrire une population dans laquelle chaque individu de la génération produit un nombre aléatoire d'individus dans la génération .
Log probabilityIn probability theory and computer science, a log probability is simply a logarithm of a probability. The use of log probabilities means representing probabilities on a logarithmic scale , instead of the standard unit interval. Since the probabilities of independent events multiply, and logarithms convert multiplication to addition, log probabilities of independent events add. Log probabilities are thus practical for computations, and have an intuitive interpretation in terms of information theory: the negative of the average log probability is the information entropy of an event.
Taylor expansions for the moments of functions of random variablesIn probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. Given and , the mean and the variance of , respectively, a Taylor expansion of the expected value of can be found via Since the second term vanishes. Also, is . Therefore, It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions.
Gilbert–Shannon–Reeds modelIn the mathematics of shuffling playing cards, the Gilbert–Shannon–Reeds model is a probability distribution on riffle shuffle permutations that has been reported to be a good match for experimentally observed outcomes of human shuffling, and that forms the basis for a recommendation that a deck of cards should be riffled seven times in order to thoroughly randomize it. It is named after the work of Edgar Gilbert, Claude Shannon, and J. Reeds, reported in a 1955 technical report by Gilbert and in a 1981 unpublished manuscript of Reeds.
Processus de Poissonvignette|Schéma expliquant le processus de Poisson Un processus de Poisson, nommé d'après le mathématicien français Siméon Denis Poisson et la loi du même nom, est un processus de comptage classique dont l'équivalent discret est la somme d'un processus de Bernoulli. C'est le plus simple et le plus utilisé des processus modélisant une . C'est un processus de Markov, et même le plus simple des processus de naissance et de mort (ici un processus de naissance pur).
Random numberIn mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness. A 1964-developed algorithm is popularly known as the Knuth shuffle or the Fisher–Yates shuffle (based on work they did in 1938). A real-world use for this is sampling water quality in a reservoir. In 1999, a new feature was added to the Pentium III: a hardware-based random number generator. It has been described as "several oscillators combine their outputs and that odd waveform is sampled asynchronously.
Variance functionIn statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function is a measure of heteroscedasticity and plays a large role in many settings of statistical modelling. It is a main ingredient in the generalized linear model framework and a tool used in non-parametric regression, semiparametric regression and functional data analysis. In parametric modeling, variance functions take on a parametric form and explicitly describe the relationship between the variance and the mean of a random quantity.
Paradoxe du singe savantLe paradoxe du singe savant est un théorème selon lequel un singe qui tape indéfiniment et au hasard sur le clavier d’une machine à écrire pourra « presque sûrement » écrire un texte donné. Dans ce contexte, « presque sûrement » est une expression mathématique ayant un sens précis, et le singe n'est pas vraiment un singe mais une métaphore pour un mécanisme abstrait qui produit une séquence aléatoire de lettres à l'infini. Le théorème illustre les dangers de raisonner sur l'infini en imaginant un très grand nombre, mais fini, et vice versa.
Algorithmically random sequenceIntuitively, an algorithmically random sequence (or random sequence) is a sequence of binary digits that appears random to any algorithm running on a (prefix-free or not) universal Turing machine. The notion can be applied analogously to sequences on any finite alphabet (e.g. decimal digits). Random sequences are key objects of study in algorithmic information theory. As different types of algorithms are sometimes considered, ranging from algorithms with specific bounds on their running time to algorithms which may ask questions of an oracle machine, there are different notions of randomness.
