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A discrete random variable, Y, follows the Poisson distribution with parameter 7s. if [13' P(Y = y; it) = e" a) Show that the log likelihood function for a sample, y = {yi 21:1, where each yi is scalar, is Tl. log L0: in) = 2H + ail-logo) town} i=1 b) Show algebraically that the Maximum Likelihood Estimator, 2, equals the sample mean, 37. Now suppose that we also observe another scalar variable xi that we believe is related to 31,-. In a Poisson regression we assume that each 37,- follows the Poisson distribution with l,- = 9.30%le The log likelihood is therefore 109 110'ny 30:31) = Z{e+lxi + J'ilo + 319%] 1090714)} i=1 c) By working with the log likelihood above, explain why Poisson regression can be thought of as an alternate version of OLS regression where the "fitted value" for each observation 1' is1 37': : eo+1xi