PoissonProcess - The Poisson Process Let > 0 A Poisson...

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1 The Poisson Process Let ! > 0 . A Poisson process with parameter distributes points along a line or in R n in a random fashion such that the distribution satisfies the following axioms. (1) Occurrences in disjoint intervals are independent. (2) In a measurable set with measure A , the probability of a single occurrence, Pr(1, A ) , satisfies: lim A ! 0 Pr(1, A ) A = " . (3) In a measurable set with measure A , the probability of n > 1 occurrences, Pr( n , A ) , satisfies: lim A ! 0 Pr( n , A ) A = 0 . As derived in class, the exact probability of n occurrences in a measurable set of measure t is exactly: p ( n ) = lim k "# k n $ % ( ) t k $ % ( ) n 1 * t k $ % ( ) k * n p ( n ) = lim k "# k ! n !( n $ k )! k n t ( ) n 1 $ t k % ( ) * k 1 $ t k % ( ) * $ n p ( n ) = lim k "# k k k $ 1 k ! k $ n + 1 k % ( ) * ( t ) n n ! % ( ) * 1 $ t
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PoissonProcess - The Poisson Process Let > 0 A Poisson...

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