[140 total points] (archaeology) Paleobotanists estimate the moment in the remote past when a given species became extinct by taking cylindrical, vertical core samples well below the earth's sur- face and looking for the last occurrence of the species in the fossil record, measured in meters above the point P at which the species was known to have first emerged. Letting {y, i-1,... ,n} denote a IID sample of such distances above P at a random set of locations, the model (Y|e) V Uniform (0,0) (*) emerges fron simple and plausible assumptions. In this model the unknown θ > 0 can be used through carbon dating, to estimate the species extinction time The marginal distribution of a single observation y in this model may be written py, (il) otherwise (12) where I(A) = 1 if A is true and 0 otherwise (a) Briefly explain why the staternent(0 Vi < θ for all i = 1, ,n) is equivalent to the statement mmax (yi, yn) S, and use this to show that the joint distribution of (Yİ, . . . ,%) in this model is 1 (m <0) θη (13) 20 points (b) Letting the observed values of (Yi, ,%) be y-y-.., yn), an important object in both frequentist ard Bayesian inferential statistics is the likelihood function 1(9 y), which is ob- tained from the joint distribution of (Yİ.. .. , Yn) simply by (1) thinking of fy, ,x(yi, . . . ,%) as a function of θ for fixed y, and (2) multiplying by an arbitrary positive constant c: (14) Using this terminology, in part (a) you showed that the likelihood function in this problem is 1(013) = θ-n1(9 > m), where m2 is the largest of the yi values. Both frequentists and Bayesians are interested in something called the mazzmum likelihood estimator (MLE) θMLE which is the value of θ that makes E(θ|y) as large as possible. (i) Make a rough sketch of the likelihood function, and use your sketch to show that the MILE in this problern is θMLE = m = max (yl, . . . yn). /20 points/ (ii) Maximization of a function is usually accomplished by setting its first derivative to 0 and solving the resulting equation. Briefly explain why that method won't work in finding the MLE in this case.