 # Random walk Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis). A random walk is a mathematical formalization of a path that consists of a succession of random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be modeled as random walks, although they may not be truly random in reality. The term random walk was first introduced by Karl Pearson in 1905. Random walks have been used in many fields: ecology, economics, psychology, computer science, physics, chemistry, and biology. Random walks explain the observed behaviors of processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. Various different types of random walks are of interest. . . Lattice random walk .

Markov chains don’t converge I often hear people often say they’re using a burn-in period in MCMC to run a Markov chain until it converges. But Markov chains don’t converge, at least not the Markov chains that are useful in MCMC. These Markov chains wander around forever exploring the domain they’re sampling from. Not only that, Markov chains can’t remember how they got where they are. When someone says a Markov chain has converged, they may mean that the chain has entered a high-probability region. Burn-in may be ineffective. Why use burn-in? So why does it matter whether you start your Markov chain in a high-probability region? Samples from Markov chains don’t converge, but averages of functions applied to these samples may converge. It’s not just a matter of imprecise language when people say a Markov chain has converged.

Markov chain A simple two-state Markov chain A Markov chain (discrete-time Markov chain or DTMC), named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another on a state space. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes. Introduction A Markov chain is a stochastic process with the Markov property. In literature, different Markov processes are designated as "Markov chains". The changes of state of the system are called transitions. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. Many other examples of Markov chains exist. Formal definition . . of states as input, where is defined, while and

Markov process Markov process example Introduction A Markov process is a stochastic model that has the Markov property. Note that there is no definitive agreement in literature on the use of some of the terms that signify special cases of Markov processes. Markov processes arise in probability and statistics in one of two ways. Markov property The general case Let , for some (totally ordered) index set ; and let be a measurable space. adapted to the filtration is said to possess the Markov property with respect to the if, for each and each with s < t, A Markov process is a stochastic process which satisfies the Markov property with respect to its natural filtration. For discrete-time Markov chains In the case where is a discrete set with the discrete sigma algebra and , this can be reformulated as follows: Examples Gambling Suppose that you start with \$10, and you wager \$1 on an unending, fair, coin toss indefinitely, or until you lose all of your money. , then the sequence .

Poisson process In probability theory, a Poisson process is a stochastic process that counts the number of events[note 1] and the time that these events occur in a given time interval. The time between each pair of consecutive events has an exponential distribution with parameter λ and each of these inter-arrival times is assumed to be independent of other inter-arrival times. The process is named after the French mathematician Siméon Denis Poisson and is a good model of radioactive decay, telephone calls and requests for a particular document on a web server, among many other phenomena. The Poisson process is a continuous-time process; the sum of a Bernoulli process can be thought of as its discrete-time counterpart. A Poisson process is a pure-birth process, the simplest example of a birth-death process. Definition Consequences of this definition include: Other types of Poisson process are described below. Types Homogeneous Sample Path of a counting Poisson process N(t) , where .

Poisson distribution In probability theory and statistics, the Poisson distribution (French pronunciation: ​[pwasɔ̃]; in English often rendered ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume. For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. Definitions Probability mass function The Poisson distribution is popular for modeling the number of times an event occurs in an interval of time or space. where The positive real number λ is equal to the expected value of X and also to its variance Example (with If .

Binomial distribution Binomial distribution for with n and k as in Pascal's triangle The probability that a ball in a Galton box with 8 layers (n = 8) ends up in the central bin (k = 4) is In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. Specification Probability mass function In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ B(n, p). for k = 0, 1, 2, ..., n, where is the binomial coefficient, hence the name of the distribution. different ways of distributing k successes in a sequence of n trials. Looking at the expression ƒ(k, n, p) as a function of k, there is a k value that maximizes it. where Example

Geometric distribution In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... } Which of these one calls "the" geometric distribution is a matter of convention and convenience. These two different geometric distributions should not be confused with each other. It’s the probability that the first occurrence of success require k number of independent trials, each with success probability p. for k = 1, 2, 3, .... The above form of geometric distribution is used for modeling the number of trials until the first success. for k = 0, 1, 2, 3, .... In either case, the sequence of probabilities is a geometric sequence. Moments and cumulants Let μ = (1 − p)/p be the expected value of Y. [citation needed]

Probability These concepts have been given an axiomatic mathematical formalization in probability theory (see probability axioms), which is used widely in such areas of study as mathematics, statistics, finance, gambling, science (in particular physics), artificial intelligence/machine learning, computer science, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems. Interpretations When dealing with experiments that are random and well-defined in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the statistical number of outcomes considered favorable divided by the total number of all outcomes (tossing a fair coin twice will yield head-head with probability 1/4, because the four outcomes head-head, head-tails, tails-head and tails-tails are equally likely to occur). Etymology History where

Handbook of Mobile Ad Hoc Networks for Mobility Models - Radhika Ranjan Roy

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