I would like to know if the law of the unconscious statistician can be applied for stochastic processes.
If so, can someone provide a reference?
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The law of the unconscious statistician is a basic theorem of probability, and it can be applied whenever you know the distribution of a random variable and want the expected value of a function of it. That does happen a lot in stochastic processes. Just about any text on stochastic processes should include tons of examples.