Transcription of LECTURE 5 - UC Davis Mathematics
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LECTURE 5 - UC Davis Mathematics
128 LECTURE 5 stochastic ProcessesWe may regard the present state of the universe as the effectof its past and the cause of its future. An intellect which at acertain moment would know all forces that set nature in motion,and all positions of all items of which nature is composed, if thisintellect were also vast enough to submit these data to analysis, itwould embrace in a single formula the movements of the greatestbodies of the universe and those of the tiniest atom; for such anintellect nothing would be uncertain and the future just like thepast would be present before its many problems that involve modeling the behavior of some system, we lacksufficiently detailed information to determine how the system behaves, or the be-havior of the system is so complicated that an exact description of it becomesirrelevant or impossible. In that case, a probabilistic model is often and randomness have many different philosophical interpretations,but, whatever interpretation one adopts, there is a clear mathematical formulationof probability in terms of measure theory, due to is an enormous field with applications in many different areas.
LECTURE 5. STOCHASTIC PROCESSES 133 We say that random variables X 1;X 2;:::X n: !R are jointly continuous if there is a joint probability density function p(x
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