By Scott M.
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Additional info for Applied stochastic processes in science and engineering
A. Gardner (McGraw-Hill, 1990). The second book (by Gardner) is particularly recommended for electrical engineering students or those wishing to use stochastic processes to model signal propagation and electrical networks. Exercises 1. White noise: It is often useful, in cases where the time-scale of the ﬂuctuations is much shorter than any characteristic time scale in the system of interest, to represent the ﬂuctuations as white noise. (a) Show that for ????-correlated ﬂuctuations, ????(???? ) = ????(???? ), the ﬂuctuation spectrum is constant.
How does this irreversibility come about? This is one of the classic problems of statistical mechanics (see M. C. , time-dependent perturbation theory in quantum mechanics). This is not enough, of course, for we need to know the evolution of the system on a time scale of the order of the time it takes to perform an experiment. It is here, namely in bridging these two time scales, that the Markovian assumption helps: from our knowledge of the transition probability at small times we can build up our knowledge of the transition probability at all time iteratively from the Chapman-Kolmogorov equation.
Intuitively, the relationship between the conditional and joint probabilities is very straightforward. Focusing upon the two variable case, ???? (????1 , ????2 ; ????1 , ????2 ) = ???? (????2 , ????2 ∣????1 , ????1 ) × ???? (????1 , ????1 ), simply means the probability that the state is ????2 at time ????2 and ????1 at ????1 is equal to the probability that the state is ????2 at time ????2 given that it was ????1 at ????1 multiplied by the probability that the state actually was ????1 at ????1 . 8) −∞ the correlation function by, ∫∞ ∫∞ ????(????1 , ????2 ) = ⟨????(????1 )????(????2 )⟩ = ????1 ????2 ???? (????1 , ????2 ; ????1 , ????2 ) ????????1 ????????2 .
Applied stochastic processes in science and engineering by Scott M.