You own a store that sells light bulbs. Customers have been complaining that the bulbs you sell burn out quickly. You want to identify which of two factories (Factory A or Factory B) is the one producing the current batch of light bulbs in your store. You have n light bulbs in your test batch, but you do not know from which factory these bulbs came. Furthermore, you have no a priori reason to think one factory is more likely than the other. You have examined m light bulbs (where m < n) and you have measured the lifetime of these m light bulbs.
Assume that the light bulbs produced in Factory A have a lifetime you can model as being exponentially distributed with parameter A. Assume that the light bulbs produced in Factory B have a lifetime you can model as being exponentially distributed with parameter B. Finally, assume that the lifetimes of the light bulbs are conditionally independent given the parameters.
Let be the exponential failure rate for light bulbs in your test batch of size n.
(a) Assuming that you know A and B, write the posterior probability distribution of .
(b) Assume the following independent prior distributions for A and B:
Aj A; A Gamma( A; A)
Bj B; B Gamma( B; B)
The posterior distribution of with these assumptions. What is the name of this type of distribution?