Stochastic

Stochastic processes

M. Veeraraghavan; Feb. 10, 2004 A stochastic process (SP) is a family of random variables { X ( t ) t ∈ T } defined on a given probability space, indexed by the time variable t , where t varies over an index set T . [1]

Just as a random variable assigns a number to each outcome s in a sample space S , a stochastic process assigns a sample function x ( t, s ) to each outcome s . 1

A sample function x ( t, s ) is the time function associated with outcome s of an experiment. The ensemble of a stochastic process (sp) is the set of all possible time functions that can result from an experiment.

x ( t, s 1 ) s1 s2 s3 x ( t, s 2 ) x ( t, s 3 )

Sample function

Figure 1:Representation of a stochastic process - relation to random variables

t = t1 X ( t 1 )is a random variable

Example of an sp: Number of active calls M ( t ) at a switch at time t . One trial of the experiment yields the sample function m ( t, s ) where the number of active calls is measured for every second over one 15 minute interval. Say this measurement is taken every day starting at 10AM. An ensemble average can be obtained from all measurements for t = 2min after 10AM. Or a time average can be obtained over a 15-minute interval based on one-day’s measurements.

1. Most of the statements in this writeup have been taken verbatim from [2]; exceptions are primary from [1] as noted.

1

Types of stochastic processes: Discrete value and continuous value; Discrete time and continuous time. If each random variable X ( t ) for different values of t are discrete rv, then the sp is a discrete value. If the process is defined only for discrete time instants, then it is a discrete time sp.

Random sequence: for a discrete time process, a random sequence X n is an ordered sequence of random variables X 0 , X 1 , ....- Essentially a random sequence is a discrete-time stochastic process. Relation between sp and rv: A discrete value sp is defined by the joint PMF P X ( t1...