Question

4. Let X be a random variable that describes the annual counts of tropical cyclones in the North Atlantic. Assume that X1,..., X, is a random sample that describes the counts of tropical cyclones in the North Atlantic during n years and assume they are distributed according to a geometric distribution with probability parameter 8 and p.f. given by fxjex | ) = (1 - 0)*-11{1,2,...,x), 0<O<1. (a) Write the statistical model. (b) Find the maximum likelihood estimator of 0. (c) Find the maximum likelihood estimator of the probability of the annual counts of tropical cy- clones in the North Atlantic to be equal to 2. (d) Show that the posterior distribution of O is a Beta distribution with shapel parameter n + 1 and shape2 parameter 1-11; - n+1, this is Eox f(-1;+2) _"(1 - 0) T(n + 1)( Ii-n+1) 3) = 12:110,1(C). Hint S 0 (1 - 0)'de = [(a+1)(6+1) (e) From a sample of size n = 2 it was observed that a = 10. Compute the maximum likelihood estimate and Bayes estimate of 0. Compare these estimates. (f) From another sample of size n = 100 it was observed that I = 10 (again). Compute the maximum likelihood estimate and Bayes estimate of 0. Compare these estimates. How can you explain the differences you observe between (d) and (e)?

Answer 1

statistical model - joint pry of Xita--yln - falo) N N on cr-o sin dog likelihood a L10) on log 01/291-1) Log (1-0) 2460 -0 = 3 [n logo + (321-9) Log (1-0)] =0 ) = + = (-) = 0 - - n-no - Orxi - no na osni 3 ô - a - © P(x=2) 01102, 061-09 Now, MLE of this probability a ocro) = 1 / 4 (1-2)

@ ocoll Assuming uniform prior onen we have, poslenior dicht bertem of 8 s. ¢ (lx) = $(210) ACO) fralo) A Col do - 0 (1-0) Erin oh Cerojening = 0 (1-0) Eairn = I ghi - Cine Sitt-na / on Cr-o) Exi- n e Blnt, Enirni) Tenitz on Como Eu i-a InH Eni-nh Hence, postenior distribution of o is Beta dish with parameters (mt) & Zairnt) n2z, n=10 MLE - Za to 2011 Bayes estimate of Oz postenis mean of a O nal . .: Eu+2 ИН nat2 Bayes estimate >ML estimate 27 - 2x10+2 2-32 -00136

mnt 2 6 n= too, n = 10 . ML estimate a To = 0.1 Bayes estimate 2 NHL 100H rooxlo)+2 with the increment in no ML remain 1002 same as long as 7 remains came bat = 0.1008 bayes estimate is getting changed But as nuo Bayes estimate becomes closer to ML estimate.