Elements of the Theory of Markov Processes and Their ApplicationsThis graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also covers mathematical models based on the theory, employed in various applied fields. Prerequisites are a knowledge of elementary probability theory, mathematical statistics, and analysis. Appendixes. Bibliographies. 1960 edition. |
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Page 20
... mean and variance of X1 , respectively . It is well known that and ∞ -Σxp ( x ) m = ¿ { X1 } = Σ xp ( x ) x = 0 o2 = 9o { X , } = Σ x2p ( x ) — m2 x = 0 - ( 1.32 ) ( 1.33 ) Let us assume that m and o2 are finite . By using the method ...
... mean and variance of X1 , respectively . It is well known that and ∞ -Σxp ( x ) m = ¿ { X1 } = Σ xp ( x ) x = 0 o2 = 9o { X , } = Σ x2p ( x ) — m2 x = 0 - ( 1.32 ) ( 1.33 ) Let us assume that m and o2 are finite . By using the method ...
Page 75
... mean and variance of the Poisson process . By definition ∞ m ( t ) = & { X ( t ) } = ΣxPx ( t ) hence ∞ m ( t ) = e − λt √x 2 x ! x = 0 ( 2t ) x x = 0 Σχ = λte - λt ( 2t ) = - 1 x = 1 ( x − 1 ) ! λte - λteit = λt ( 2.94 ) Hence , the ...
... mean and variance of the Poisson process . By definition ∞ m ( t ) = & { X ( t ) } = ΣxPx ( t ) hence ∞ m ( t ) = e − λt √x 2 x ! x = 0 ( 2t ) x x = 0 Σχ = λte - λt ( 2t ) = - 1 x = 1 ( x − 1 ) ! λte - λteit = λt ( 2.94 ) Hence , the ...
Page 76
... mean and variance are equal . Thus far we have considered only the function P ( t ) , which has been interpreted as the probability that exactly x changes have taken place in an interval of length t . Let us now consider the transition ...
... mean and variance are equal . Thus far we have considered only the function P ( t ) , which has been interpreted as the probability that exactly x changes have taken place in an interval of length t . Let us now consider the transition ...
Page 78
... mean of X ( t ) , we have 00 m ( t ) = & { X ( t ) } = ΣxPx ( t ) Σ xP2 ( t ) = e − 1t Σ x ( 1 x = 0 · e ̄1t Σ x ... variance is D2 { X ( t ) } = e1t ( eåt — 1 ) ( 2.111 ) Before ending our discussion of the birth process , we 78 THEORY ...
... mean of X ( t ) , we have 00 m ( t ) = & { X ( t ) } = ΣxPx ( t ) Σ xP2 ( t ) = e − 1t Σ x ( 1 x = 0 · e ̄1t Σ x ... variance is D2 { X ( t ) } = e1t ( eåt — 1 ) ( 2.111 ) Before ending our discussion of the birth process , we 78 THEORY ...
Page 80
... mean and variance in this case are given by and m ( t ) = & { X ( t ) } = ie1t D2 { X ( t ) } = iet ( eλt — 1 ) - ( 2.117 ) ( 2.118 ) In the general birth process the birth rate depends on. Before ending our discussion of the birth ...
... mean and variance in this case are given by and m ( t ) = & { X ( t ) } = ie1t D2 { X ( t ) } = iet ( eλt — 1 ) - ( 2.117 ) ( 2.118 ) In the general birth process the birth rate depends on. Before ending our discussion of the birth ...
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Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |
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