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IEOR 165 Lecture 4 June 2 2009

# IEOR 165 Lecture 4 June 2 2009 - Lecture Notes IEOR 165 |...

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Lecture Notes IEOR 165 | George Shanthikumar Tu W Th 2-4:30 pm | 3113 Etcheverry We are given a population X. We want the mean , = - . μ variance σ2 Ex2 Ex2 x is an estimator of μ S2is an estimator of σ2 Suppose we know more about the population? Can we use this knowledge? The above is applicable regardless of the probability distribution function (PDF) of x. Suppose we know the parametric family of the PDF of x. (e.g. binomial distribution? Poisson? Etc) Example : Suppose x has an exponential distribution function. = - , ≥ fxx λe λx x 0 . Mathematically, it seems like we know everything we want. But Actually, we can’t compute because we don’t know Lambda. Therefore, to fully characterize x, we need to estimate . λ (parameter estimation) Consider = ∞ - = Ex 0 xλe λxdx : - <- - . . hint 1λ0 λxe λxλdx density function of Erlang 2 r v = ∞ - = Ex2 0 x2λe λxdx 2λ2 : - ! <- - . . hint 2λ20 λx2e λx2 λdx density function of Erlang 3 r v So = - = Varx 2λ2 1λ2 1λ2 , = = Therefore Ex and λ 1Ex We don’t know , λ or Ex so we need to estimate: =

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IEOR 165 Lecture 4 June 2 2009 - Lecture Notes IEOR 165 |...

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