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Unformatted text preview: Homework 4, Math 3225, Fall 2010 November 22, 2010 1. In this problem I will walk you through a proof that the sample variance is unbiased. Let S 2 = ∑ n i =1 ( X i X ) 2 n 1 denote the sample variance, where we assume X 1 , ..., X n are i.i.d. a. First show that E ( S 2 ) = n ( E ( X 2 1 ) E ( X 2 )) n 1 . b. Next, using the fact that V ( X ) = E ( X 2 ) E ( X ) 2 , show that E ( X 2 ) = V ( X 1 ) /n + E ( X 1 ) 2 . c. Finally, deduce that S 2 is unbiased. 2. Determine, with proof, a maximum likelihood estimate for the param eter λ in a Poisson random sample. That is to say, given a sample X 1 , ..., X k of i.i.d. random variables sampled from a Poisson distribu tion having pdf λ n e λ /n !, determine the maximum likelihood estimator ˆ λ = ˆ λ ( X 1 , ..., X k ). 3. Recall that the sample correlation coefficient r for a sample ( X 1 , Y 1 ) , ..., ( X n , Y n ) is defined to be r = S XY √ S XX √ S Y Y , where for sample variables ( V i , W i ) we define S V W := summationdisplay...
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 Spring '08
 Staff
 Statistics, Normal Distribution, Probability, Variance, maximum likelihood estimator, maximum likelihood estimate, sample variance, λn e−λ /n, Poisson random sample

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