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Unformatted text preview: RANDOM VARIABLES so the probability mass at zero for the binomial distribution converges to the probability mass at
zero for the Poisson distribution. Similarly, for any integer k ≥ 0 ﬁxed,
p b (k ) = nk
p (1 − p)n−k
k
k = n · (n − 1) · · · (n − k + 1)
k! = λk pb (0) n · (n − 1) · · · (n − k + 1)
k!
nk → λk e−λ
k! λ
n 1− λ
n n−k λ
1−
n −k (2.6) as n → ∞, because (2.5) holds, and the terms in square brackets in (2.6) converge to one.
Checking that the Poisson distribution sums to one, and deriving the mean and variance of the
Poisson distribution, can be done using the pmf as can be done for the binomial distribution. This
is not surprising, given that the Poisson distribution is a limiting form of the binomial distribution.
The MacLaurin series for ex plays a role:
∞
x e=
k=0 xk
.
k! (2.7) Letting x = λ, and dividing both sides of (2.8) by eλ , yields
∞ 1=
k=0 λk e−λ
,
k! (2.8) so the pmf does sum to one. Following (2.4) line for line yields that if Y has the Poisson distribution
with param...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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