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Unformatted text preview: n t} be the number of arrivals by time t, which is Poisson( t). N (t) has 92 CHAPTER 2. POISSON PROCESSES independent increments: if t0 < t1 < . . . < tn , then N (t1 )
N (t1 ), . . . N (tn ) N (tn 1 ) are independent. N (t0 ), N (t2 ) Thinning. Suppose we embellish our Poisson process by associating to each
arrival an independent and identically distributed (i.i.d.) positive integer random variable Yi . If we let pk = P (Yi = k ) and let Nk (t) be the number of
i N (t) with Yi = k then N1 (t), N2 (t), . . . are independent Poisson processes
and Nk (t) has rate pk .
Random sums. Let Y1 , Y2 , . . . be i.i.d., let N be an independent nonnegative
integer valued random variable, and let S = Y1 + · · · + YN with S = 0 when
N = 0.
(i) If E |Yi |, EN < 1, then ES = EN · EYi . (ii) If EYi2 , EN 2 < 1, then var (S ) = EN var (Yi ) + var (N )(EYi )2 .
(iii) If N is Poisson( ) var (S ) = E (Yi2 ) Superposition. If N1 (t) and N2 (t) are independent Poison processes with
rates 1 and 2 then N1 (t) + N2 (t) is Poisson rate 1 + 2 .
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