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Example 11.6

Example 11.6 - 11.2 RANDOM PROCESSES WITH EVOLUTIONARY...

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11.2. RANDOM PROCESSES WITH EVOLUTIONARY DESCRIPTIONS 355 Using the Laplace transform L { F (0 , T ) } = L { e - λT } = 1 / ( s + λ ), It follows that L { F ( k, T ) } = λ s + λ k 1 s + λ L - 1 = F ( k, T ) = ( λT ) k k ! e - λT Hence the distribution of the number of points in an interval of duration T is Poisson with parameter λT . The parameter λ is called the rate ( event rate or point rate ) because the average number of points per second is E { N ( u, ( a, a + T ] ) } T = λ . With attribute (c) and the above Poisson probability function, it is possible to write down the joint probability of the numbers of points in any finite number of disjoint intervals in R , and often this will be enough to answer most questions of interest. A random process often associated with the basic Poisson point process is the Poisson counting process x ( u, t ) , N ( u, (0 , t ] ), t [0 , ), with P { x ( u, 0) = 0 } = 1. All sample functions of x ( u, t ) must be monotone increasing “staircase functions”, and hence for t 1 < t 2 < . . . < t m , P { x ( u, t 1 ) = k 1 , x ( u, t 2 ) = k 2 , . . . , x ( u, t m ) = k m } = F ( k 1 , t 1 ) m Y n =2 F ( k n - k n - 1 , t n - t n - 1 ) , k 1 k 2 . . . k m 0 , otherwise. This specification of the joint probability of an arbitrary finite collection of random variables in the random process x ( u, t ), t [0 , ), constitutes a complete description of the Poisson counting process x ( u, t ). The name “Poisson” logically can be attached to any random process with associated random variables that are distributed according to a Poisson distribution. It should be obvious to the reader that the Poisson counting process x ( u, t ) is also a Markov process (why?) and a second-order process (why?).
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