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Unformatted text preview: · · + YN (t) where we set S (t) = 0 if N (t) = 0. In Example 2.1, S (t) gives the number of
customers that have arrived up to time t. In Example 2.2, S (t) represents the
total number of bytes in all of the messages up to time t. In each case it is
interesting to know the mean and variance of S (t).
Theorem 2.10. Let Y1 , Y2 , . . . be independent and identically distributed, 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( ), then var (S ) = E Yi2 . 87 2.3. COMPOUND POISSON PROCESSES Why is this reasonable? The ﬁrst of these is natural since if N = n is
nonrandom ES = nEYi . (i) then results by setting n = EN . The formula in
(ii) is more complicated but it clearly has two of the necessary properties:
If N = n is nonrandom, var (S ) = n var (Yi ).
If Yi = c is nonrandom var (S ) = c2 var (N ).
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- Spring '10
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