A river where drinking water is taken is feared to have been polluted by coliform bacteria In order to investigate this, we take a

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A river where drinking water is taken is feared to have been polluted by coliform bacteria In order to investigate this, we take a sample tube of water at n randomly selected sites from the river tube and from each of these samples we determine the numbers of coliform bacteria Y1,..., Yn. A classical distribution used to describe the distribution of the number of bacteria per unit volume of water, is the Poisson distribution. So we model our measurement results so that they form a random sample Y,..., Yn from Poisson distribution, i.e. we assume we have n independent observations from the Poisson distribution. (a) Construct the likelihood function L(\; y) of the model and construct the log- likelihood function ((\; y) of the model (b) Derive by examining the log-likelihood function and carefully justifying the max- imum likelihood estimate of the parameter \ (c) Show with justifications that the ML estimator of the model is unbiased (d) Calculate the mean squared error msex (^(Y)) of the ML estimator of the model