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Figure 1. An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at [0,0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. stable) distribution with α = 1 and β = 0 which is a Cauchy distribution. Note the presence of large jumps in location compared ...
A Bernoulli process is a finite or infinite sequence of independent random variables X 1, X 2, X 3, ..., such that . for each i, the value of X i is either 0 or 1;; for all values of , the probability p that X i = 1 is the same.
Five eight-step random walks from a central point. Some paths appear shorter than eight steps where the route has doubled back on itself. (animated version)In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some mathematical space.
In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times.
Strictly speaking, the EMC is a regular discrete-time Markov chain. Each element of the one-step transition probability matrix of the EMC, S, is denoted by s ij, and represents the conditional probability of transitioning from state i into state j. These conditional probabilities may be found by
The most common formulation of a branching process is that of the Galton–Watson process.Let Z n denote the state in period n (often interpreted as the size of generation n), and let X n,i be a random variable denoting the number of direct successors of member i in period n, where X n,i are independent and identically distributed random variables over all n ∈{ 0, 1, 2, ...} and i ∈ {1 ...
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.
A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Q k for all k (from 0 to ∞).