The lifetime of a type-A bulb is exponentially distributed with parameter λ. The lifetime of a type-B bulb is exponentially distributed with parameter μ, where μ>λ>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 μ≥3λ. You observe the value t 1 of the lifetime, T1, of a lightbulb. A MAP decision rule decides that the lightbulb is of type A if and only if t 1 ≥α. Find α, and express your answer in terms of μ and λ. Use ‘mu" and ‘lambda" and ‘ln" to denote μ, λ, and the natural logarithm function, respectively. For example, ln 2μ λ should be entered as ‘ln((2*mu)/lambda)".
2) Assume again that μ≥3λ. What is the probability of error of the MAP decision rule?

a) 1/4 e −μα + 3 4 (1− e −λα )
b) 3/4 e −μα + 1 4 (1− e −λα )
c) 1/4 (1− e −μα )+ 3 4 e −λα
d) 3/ 4 (1− e −μα )+ 1 4 e −λα

3) Assume that λ=3 and μ=4. Find the LMS estimate of T2, the lifetime of another lightbulb from the same box, based on observing T1=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.)