A lifetime X (in years) of a particular machine is modeled by an exponential distribution with unknown failure rate parameter 6'. The lifetimes of X I = 5. X 2 = 6. and X 3 = 4 are observed. and...

A lifetime X (in years) of a particular machine is modeled by an exponential distribution with unknown failure rate parameter 6'. The lifetimes of X_I
= 5. X₂
= 6. and X₃
= 4 are observed. and assume that an expert believes that 6' should have exponential distribution as well and that, on average
 should be 1/3.

(i) VL-rite down the lILE of 6' for those observations.

(ii) Elicit a prior according to the expert's beliefs.

(iii) For the prior in (ii). find the posterior. Is the problem conjugate?

(iv) Find the Baves estimator 8_Boyes. and compare it with the LlLE estimator from (i). Discuss.