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Unformatted text preview: eter λ,
∞ k λ k e −λ
k! k E [Y ] = λk e−λ
k! k=0
∞ =
k=1
∞ =λ
k=1
∞ =λ
l=0 λk−1 e−λ
(k − 1)!
λl e−λ
l! (here l = k − 1) = λ.
Similarly, it can be shown that Var(Y ) = λ. The mean and variance can be obtained by taking
the limit of the mean and limit of the variance of the binomial distribution with parameters n and 2.8. MAXIMUM LIKELIHOOD PARAMETER ESTIMATION 43 λ
p = n/λ, as n → ∞, as follows. The mean of the Poisson distribution is limn→∞ n n = λ, and the
λ
λ
variance of the Poisson distribution is limn→∞ n n 1 − n = λ. Example 2.7.1 It is assumed that X has a Poisson distribution with some parameter λ with
λ ≥ 0, but the value of λ is unknown. Suppose it is observed that X = k for a particular integer
k. Find the maximum likelihood estimate of X.
k −λ e
Solution: The likelihood of observing X = k is λ k! ; the value of λ maximizing such likelihood
is to be found for k ﬁxed. Equivalently, the value of λ maximizing λk e−λ is to be found. If k ≥ 1, d(λk e−λ )
=...
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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 Tech.
 Spring '08
 Zahrn
 The Land

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