Random Process

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Random or Stochastic Processes

A #~{random process} ( or #~{stochastic process} ) is one which develops over time or space. Mathematically it is a collection of random variables, either:

  • \{ ~X ( ~t ) ; ~t &in. &reals. \} or \{ ~X ( ~t ) ; ~t > 0 \} is a #~{random process in continuous time}.
  • \{ ~X_~n ; ~n = 0, 1, 2, ... \} is a #~{discrete-time random process}.

The variables ~X ( ~t ) 0r ~X_~n can be either discrete or continuous:

  • ~X discrete: the process is called a #~{discrete-valued random process}.
  • ~X continuous: the process is called a #~{continuous-valued random process}.

So there are four possible main categories of random processes, here are some examples of each:

Discrete-valued Discrete-Time Random Processes

  1. A coin is tossed a number of times. If on the ~n^{th} toss the result is a "head" then ~X_~n = 1, if it is a "tail" then ~X_~n = 0 , ~n = 1, 2, ... . This is an example of a Bernoulli process .
  2. The number of 'hits' on a web-site each day is measured, ~X_~n being simply the number of hits on day ~n.

Discrete-valued Random Processes in Continuous Time

  1. poisson.

Continuous-valued Discrete-Time Random Processes

  1. Daily rainfall is measured at a particular spot over a period of time, ~X_~n is the amount of rainfall (in millimeters) measured on day ~n.

Continuous-valued Random Processes in Continuous Time

  1. A record of temperature is kept. Temparature (measured in &degree.C) is a continuous variable, and it can be measured at any point in time. In practice of course, temperature will be measured at regular intervals for meterological records, making it a discrete-time process, but in theory this is a process in continuous time, ~X(~t) being the temperature at time ~t.