Suppose that a scientist desires to estimate the proportion? of monarch butterflies with a special type of marking on their wings. Suppose he captures monarch butterflies one at a time, and at the end of the day the scientist had captured 60 monarch butterflies and had found only two with special marking.
(a) Show that the family of beta-distributions constitutes a conjugate prior family for this model
(b) Based on some prior research one assumes a beta-prior for ? with parameters a = 1 and b = 9. What is the Bayes estimator of p with respect to quadratic loss?
(c) What is the MLE of ??