Let X1, ..., Xn be i.i.d Poisson(λ). a) Find the maximum likelihood estimator of λ. b) The number of surface imperfections for a random sample of 50 metal plates are summarized in the following table: Number of Scratches per Item 0 1 2 3 4 Frequency 4 12 11 14 9 Assuming that the imperfection counts have the Poisson(λ) distribution, compute the maximum likelihood estimate of λ. c) Give the model-based estimate of the population variance, and compare it with the sample variance. Assuming the Poisson model correctly describes the population distribution, which of the two estimates would you prefer and why?
