Random Processes

           In practical problems, we deal with time varying waveforms whose value at a time is random in nature. For example, the speech waveform recorded by a microphone, the signal received by communication receiver or the daily record of stock-market data represents random variables that change with time. How do we characterize such data? Such data are characterized as random or stochastic processes. This lecture covers the fundamentals of random processes.

Random processes

           Recall that a random variable maps each sample point in the sample space to a point in the real line. A random process maps each sample point to a waveform.

           Consider a probability space . A random process can be defined on as an indexed family of random variables where is an index set, which may be discrete or continuous, usually denoting time. Thus a random process is a function of the sample point and index variable and may be written as .

Remark

• For a fixed     is a random variable.