We assumed that 1 k n 1 but if k 0 then the likelihood

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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.

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