Suppose that the time to complete a task is the sum of two parts X and Y. Let (Xi, Yi) for i = 1, . . . , n be a random sample of the times to complete the two parts of the task. However, for some observations, we get to observe only Zi = Xi + Yi. To be precise, suppose that we observe (Xi, Yi) for i = 1, . . . , k and we observe Zi for i = k + 1, . . . , n. Suppose that all Xi and Yj are independent with every Xi having the exponential distribution with parameter λ and every Yj having the exponential distribution with parameter μ.
a. Prove that the conditional distribution of xi given Zi = z has the c.d.f. G(x|z) = 1− exp(−x[λ − μ]) 1− exp(−z[λ − μ]) , for 0 < x < z.
b. Suppose that the prior distribution of (λ, μ) is as follows: The two parameters are independent with λ having the gamma distribution with parameters a and b, and μ having the gamma distribution with parameters c and d. The four numbers a, b, c, and d are all known constants. Set up a Gibbs sampling algorithm that allows us to simulate the posterior distribution of (λ, μ).