Exponential gamma model. Let xi Exp(), i = 1, . . . , n, i.i.d. a. Assuming is distributed as Ga(, ),

Exponential ∧ gamma model. Let xi ∼ Exp(θ), i = 1, . . . , n, i.i.d.

a. Assuming θ is distributed as Ga(α, β), with α and β fixed, find the posterior distribution h(θ | x1, . . . , xn).

b. Under the same assumptions, compute the predictive distribution p(xn+1 |

x1, . . . , xn).

c. Consider now a slightly different situation. Imagine that the exponential random variables represent waiting times for the recurrence of disease in n patients surgically treated at time 0. We are at time

a, and all patients are still alive. Some of the n patients may have experienced recurrence. We record their recurrence time xi (assume patients are not followed up after the recurrence, i.e., we have no further data). Other patients may still be healthy, so we don’t know exactly what the recurrence time is. We only know xi >

a. Repeat steps

(a) and

(b) for this case.