Times to Failure

Times to Failure. Three devices are monitored until failure. The observed lifetimes are 0.9, 1.8, and 0.3 years. If the lifetimes ate modeled as exponential distribution with rate Ti~ Exp (1), f (t| 1) = de-At, t> 0,> > 0. Assume exponential prior on 1, A~ Exp ( 2), T () ) = 2e-24, > > 0. (a) Find the posterior distribution of 1 (b) Find the Bayes estimator for 1 (c) Find the MAP estimator for A (d) Approximate the posterior median of 1? (e) Numerically find 95% HPD confidence interval for A (f) Numerically find 95% equitailed confidence interval for A (g) Find the posterior probability of hypothesis Ho : 1 < 1/2.