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Branching Process
In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables. The random variables of a stochastic process are indexed by the natural numbers. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation n produces some random number of individuals in generation n+1, according, in the simplest case, to a fixed probability distribution that does not vary from individual to individual. Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates 0, 1, or 2 offspring with some probability in a single time unit. Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor. A central question in the the ... [...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] |
Independent And Identically Distributed Random Variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruss–Duerinckx Theorem
The theorem of the envelopment of societies for resource-dependent populations, also called the Bruss–Duerinckx theorem, is a mathematical result on the behavior of populations which choose their society form according to only two hypotheses, namely those which are seen as most "natural": * Hypothesis 1 (H1): Individuals want to survive and see a future for their descendants, * Hypothesis 2 (H2): The average individual prefers a higher standard of living to a lower one, where H1 is supposed to precede H2 in the case of incompatibility of H1 with H2. Here populations with a society structure are modeled by so-called resource-dependent branching processes (RDBPs). The objective of RDBPs is to model different society structures and to compare the advantages and disadvantages of different societies, with the focus being on human societies. A RDBP is a discrete time branching process (BP) in which individuals are supposed to have to work in order to be able to live and to reproduce. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resource-dependent Branching Process
A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population. Here population is meant in a general sense, including a human population, animal populations, bacteria and others which reproduce in a biological sense, cascade process, or particles which split in a physical sense, and others. Members of a BP-population are called individuals, or particles. If the times of reproductions are discrete (usually denoted by 1,2, ...) then the totality of individuals present at time and living to time excluded are thought of as forming the generation. Simple BPs are defined by an initial state (number of individuals at time 0) and a law of reproduction, usually denoted by . A resource-dependent branching process (RDBP) is a discrete-time BP which models the development of a population in which individuals are supposed to have to work in order to be able to live and to reproduce. The population decides on a society form which det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branching Random Walk
In probability theory, a branching random walk is a stochastic process that generalizes both the concept of a random walk and of a branching process. At every generation (a point of discrete time), a branching random walk's value is a set of elements that are located in some linear space, such as the real line. Each element of a given generation can have several descendants in the next generation. The location of any descendant is the sum of its parent's location and a random variable. This process is a spatial expansion of the Galton–Watson process. Its continuous equivalent is called branching Brownian motion. Example An example of branching random walk can be constructed where the branching process generates exactly two descendants for each element, a ''binary ''branching random walk. Given the initial condition that ''X''ϵ = 0, we suppose that ''X''1 and ''X''2 are the two children of ''X''ϵ. Further, we suppose that they are independent Independent or I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Tree
In mathematics and computer science, a random tree is a tree or arborescence that is formed by a stochastic process. Types of random trees include: *Uniform spanning tree, a spanning tree of a given graph in which each different tree is equally likely to be selected *Random minimal spanning tree, spanning trees of a graph formed by choosing random edge weights and using the minimum spanning tree for those weights *Random binary tree, binary trees with a given number of nodes, formed by inserting the nodes in a random order or by selecting all possible trees uniformly at random *Random recursive tree, increasingly labelled trees, which can be generated using a simple stochastic growth rule. *Treap or randomized binary search tree, a data structure that uses random choices to simulate a random binary tree for non-random update sequences * Rapidly exploring random tree, a fractal space-filling pattern used as a data structure for searching high-dimensional spaces *Brownian tree, a frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superprocess
An (\xi,d,\beta)-superprocess, X(t,dx), within mathematics probability theory is a stochastic process on \mathbb \times \mathbb^d that is usually constructed as a special limit of near-critical branching diffusions. Scaling limit of a discrete branching process Simplest setting For any integer N\geq 1, consider a branching Brownian process Y^N(t,dx) defined as follows: * Start at t=0 with N independent particles distributed according to a probability distribution \mu. * Each particle independently move according to a Brownian motion. * Each particle independently dies with rate N. * When a particle dies, with probability 1/2 it gives birth to two offspring in the same location. The notation Y^N(t,dx) means should be interpreted as: at each time t, the number of particles in a set A\subset \mathbb is Y^N(t,A). In other words, Y is a measure-valued random process. Now, define a renormalized process: X^N(t,dx):=\fracY^N(t,dx) Then the finite-dimensional distributions o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resource-dependent Branching Process
A branching process (BP) (see e.g. Jagers (1975)) is a mathematical model to describe the development of a population. Here population is meant in a general sense, including a human population, animal populations, bacteria and others which reproduce in a biological sense, cascade process, or particles which split in a physical sense, and others. Members of a BP-population are called individuals, or particles. If the times of reproductions are discrete (usually denoted by 1,2, ...) then the totality of individuals present at time and living to time excluded are thought of as forming the generation. Simple BPs are defined by an initial state (number of individuals at time 0) and a law of reproduction, usually denoted by . A resource-dependent branching process (RDBP) is a discrete-time BP which models the development of a population in which individuals are supposed to have to work in order to be able to live and to reproduce. The population decides on a society form which det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Generating Function
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(''X'' = ''i'') in the probability mass function for a random variable ''X'', and to make available the well-developed theory of power series with non-negative coefficients. Definition Univariate case If ''X'' is a discrete random variable taking values in the non-negative integers , then the ''probability generating function'' of ''X'' is defined as http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf :G(z) = \operatorname (z^X) = \sum_^p(x)z^x, where ''p'' is the probability mass function of ''X''. Note that the subscripted notations ''G''''X'' and ''pX'' are often used to emphasize that these pertain to a particular random variable ''X'', and to its distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term ''random walk'' was first introduced by Karl Pearson in 1905. Lattice random walk A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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