Since the events bijk involve numbers of counts in

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Unformatted text preview: . each t ≥ 0 is the cumulative number of counts up to time t. The random variables T1 , T2 , . . . are called the count times and the random variables U1 , U2 , . . . are called the intercount times. The following equations clearly hold: ∞ N (t) = I{t≥Tn } n=1 Tn = min{t : N (t) ≥ n} Tn = U1 + · · · + Un . 3.5. POISSON PROCESSES 83 Definition 3.5.1 Let λ ≥ 0. A Poisson process with rate λ is a random counting process N = (Nt : t ≥ 0) such that N.1 N has independent increments: if 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn then the increments N (t1 ) − N (t0 ), N (t2 ) − N (t1 ), . . . , N (tn ) − N (tn−1 ) are independent. N.2 The increment N (t) − N (s) has the P oi(λ(t − s)) distribution for t ≥ s. Proposition 3.5.2 Let N be a random counting process and let λ > 0. The following are equivalent: (a) N is a Poisson process with rate λ. (b) The intercount times U1 , U2 , . . . are mutually independent, exponentially distributed random variables with parameter λ. Proof. Eit...
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