Stochastic

# 23 compound poisson processes in this section we will

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Unformatted text preview: ngth of the interval, so the arrival rate of customers is constant during the hour. In reality there is a large inﬂux of people between 11:30 and 11:45 soon after the end of 10:10-11:25 classes. (i) is a very strong assumption but can be weakened by using a more general Poisson approximation result like the following: Theorem 2.9. Let Xn,m , 1 m n be independent random variables with P (Xm = 1) = pm and P (Xm = 0) = 1 pm . Let Sn = X 1 + · · · + X n , n = ESn = p1 + · · · + pn , 85 2.2. DEFINING THE POISSON PROCESS and Zn = Poisson( n ). Then for any set A |P (Sn 2 A) P (Zn 2 A)| n X p2 m m=1 Why is this true? If X and Y are integer valued random variables then for any set A 1X |P (X 2 A) P (Y 2 A)| |P (X = n) P (Y = n)| 2n The right-hand side is called the total variation distance between the two distributions and is denoted kX Y k. If P (X = 1) = p, P (X = 0) = 1 p, and Y = Poisson(p) then X |P (X = n) P (Y = n)| = |(1 p) e p | + |p pe p | + 1 (1 + p)e p n Since 1 p e e p 1 p the right-hand side is 1+p+p pe +1 p e p p pe = 2p(...
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