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
- 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 .
- 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
- poisson.
Continuous-valued Discrete-Time Random Processes
- 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
- A record of temperature is kept. Temparature (measured in °ree.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.