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 Δ*t*, 2Δ*t*, 3Δ*t*, … and use a normalized reconstruction function, , to estimate the probability density function at other times according to

(8.1)

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

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