3. Consider the following distributions and determine whether they can be put in exponential family form. If it is not possible, consider whether there is a special case which follows the form. . The Uniform distribution: f(ya, B) = B-a I(y E [a, B)) The Weibull distribution: () > 0) f(va, B) = aB aya-e(/B)" 171 The Gumbel distribution: (y > 0) 172 f(la, B) = exp ((y -a)/B-exp((x -a)/B)) 4. Consider the Galapagos data and model analyzed in this chapter. The purpose of this question is to reproduce the details of the GLM fitting of this data. (a) Fit a Poisson model to the species response with the five geographic variables as predictors. Do not use the endemics variable. Report the values of the coefficients and the deviance. (b) For a Poisson GLM, derive n, d/du, V() and the weights to be used in an iteratively fit GLM. What is the form of the adjusted dependent variable here? (c) Using the observed response as initial values, compute the first stage of the iteration, stopping after the first linear model fit. Compare the coefficients of this linear model to those found in the GLM fit. How close are they? (d) Continue the iteration to get the nextn and u. Use this to compute the current value of the deviance. How close is this to the deviance from the GLM? (e) Compute one more iteration of the GLM fit, reporting the next calculation of the coefficients and deviance. How close are these to target now? Repeat these iterations a few more times, computing the deviance in each time. Stop when the deviance does not change much. Compare your final estimated coefficients to that produced by the GLM fit. (g) Use your final iterated linear model fit to produce standard errors for the