A stokhazesthai (stochastic) process, also called a random process, is one in which outcomes are uncertain (MAT 455, ISU).
In its most general expression, a stochastic process is simply a collection of random variables {Xt, t ∈ I}. The index t often represents time, and the set I is the index set of the process. The most common index sets are I = {0, 1, 2,…}, representing discrete time, and I = [0, ∞), representing continuous time. Discrete-time stochastic processes are sequences of random variables. Continuous-time processes are uncountable collections of random variables. The random variables of a stochastic process take values in a common state space S, either discrete or continuous. A stochastic process is specified by its index and state spaces, and by the dependency relations among its random variables.
The word “stochastic” comes from the Greek stokhazesthai, which means to aim at, or guess at. A stochastic process, also called a random process, is simply one in which outcomes are uncertain. By contrast, in a deterministic system there is no randomness. In a deterministic system, the same output is always produced from a given input.
The dynamics of a stochastic process are described by random variables and probability distributions. In the deterministic growth model, one can say with certainty how many bacteria are present after t minutes.
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For the stochastic model, questions of interest might include:
• What is the average number of bacteria present at time t?
• What is the probability that the number of bacteria will exceed some threshold after t minutes?
• What is the distribution of the time it takes for the number of bacteria to double in size?
A Poisson process is a special type of counting process. Given a stream of events that arrive at random times starting at t = 0, let Nt denote the number of arrivals that occur by time t, that is, the number of events in [0, t]. For instance, Nt might be the number of text messages received up to time t. For each t ≥ 0, Nt is a random variable. The collection of random variables (Nt)t≥0 is a continuous-time, integer-valued stochastic process, called a counting process. Since Nt counts events in [0, t], as t increases, the number of events Nt increases.