Topic: Bayesian inference. Estimating the parameter of a geometric r.v.

Problem 6. Determining the type of a lightbulb 3 points possible (graded) The lifetime of a type-A bulb is exponentially distributed with parameter A. The lifetime of a type-B bulb is exponentially distributed with parameter /, where / > > > 0. You have a box full of lightbulbs of the same type, and you would like to know whether they are of type A or B. Assume an a priori probability of 1/4 that the box contains type-B lightbulbs. 1. Assume that / 2 31. You observe the value to of the lifetime, 71, of a lightbulb. A MAP decision rule decides that the lightbulb is of type A if and only if t1 2 Q. Find a, and express your answer in terms of / and ). Use 'mu" and "lambda" and 'In" to denote /, >, and the natural logarithm function, respectively. For example, In -- should be entered as 'In((2*mu)/lambda)". 2. Assume again that / 2 3). What is the probability of error of the MAP decision rule? 01 4 O 3 O 1( 1-2 40)+ 3 O ( 1 - e # )+ 3. Assume that A = 3 and / = 4. Find the LMS estimate of 72, the lifetime of another lightbulb from the same box, based on observing 71 = 2. Assume that conditioned on the bulb type, bulb lifetimes are independent. (For this part, you will need a calculator. Provide an answer with an accuracy of three decimal places.) LMS estimate of 72