4. [28 marks] In a Bayesian framework, the probability of a defect o E (0, 1) of a certain piece of a machine's output appears to be varying on various days and can be represented as a random variable with prior density T(0) = 1202(1 - 0), 0 E (0, 1). On a certain day, n = 16 observations denoted X = (X1, X2, ..., Xis) are made where 1 if the piece is defective X 0 if the piece is nondefective Hence f(X10) = 2-* (1 -0)"-2-* a) Find the formula for the Bayesian estimator (with respect to quadratic loss) of the probability o and calculate its estimated value if from the 16 pieces examined on this day, exactly three were defective. Hint: You may use: B(a, 8) = fro-'(1 -x)-'dx, B(a, P) = Total ' ror(s) T(a) = fo exp(-x)ro-'dx, ra + 1) = al(a). b) Find the Bayesian estimator (with respect to quadratic loss ) of the pa- rameter n(0) = e(1 - 0). c) Using the data of part (a) (3 defective out of 16 tested), test the hypoth- esis Ho : 0 0.25 using the Bayesian approach and a 0-1 loss. Do you accept Ho? (You may use: Jo-25 05(1 - 0)'4de = 0.00000164)