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 10:1011:25 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 righthand 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 righthand side is 1+p+p pe +1 p e p p pe = 2p(...
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 Spring '10
 DURRETT
 The Land

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