This characterization of tr and the method of section

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Unformatted text preview: her (a) or (b) provides a specific probabilistic description of a random counting process. Furthermore, both descriptions carry over as limits from the Bernoulli random counting process, as h, p → 0 with p/h → λ. This provides a proof of the proposition. Example 3.5.3 Consider a Poisson process on the interval [0, T ] with rate λ > 0, and let 0 < τ < T . Define N1 to be the number of counts during [0, τ ], N2 to be the number of counts during [τ, T ], and N to be the total number of counts during [0, T ]. Let i, j, n be nonnegative integers such that n = i + j . Express the following probabilities in terms of n, i, j, τ, T , and λ, simplifying your answers as much as possible: (a) P {N = n}, (b) P {N1 = i}, (c) P {N2 = j }, (d) P (N1 = i|N = n), (e) P (N = n|N1 = i). Solution: (a) P {N = n} = (b) P {N1 = i} = (c) P {N2 = j } = (d) e−λT (λT )n . n! e−λτ (λτ )i . i! e−λ(T −τ ) (λ(T −τ ))j . j! P (N1 = i|N = n) = = = P {N1 = i, N = n} P {N1 = i, N2 = j } = P {N = n} P {N = n} P {N1 = i}P {N2 = j } n! τ =...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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