At a set time, a random process defines a random variable that has a probability mass function, or a probability density function, as appropriate. It is of interest to be able to determine this probability mass/density function for all times—resulting in the probability mass/density function evolution with time. However, in general, such evolution is analytically difficult to establish. This evolution can be established for a few important random processes including the random walk, one-dimensional Brownian motion, and the case of random processes defined as a sum of independent random variables at each time instant.

Probability density function evolution is consistent with probability flow, and for continuous time Markov random processes, the probability density function evolution leads to the Fokker–Planck equation.

When an analytic solution cannot be obtained, estimation techniques for the probability mass/density function can be used, and this chapter commences with a brief introduction. For the continuous case, a useful approach is to estimate the probability density function at set times Δt, 2Δt, 3Δt, … and use a normalized reconstruction function, images, to estimate the probability density function at other times according to


Here, is an estimate of the probability density function of the random process X at the time ...

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