7. The exponential regression model. The exponential regression model is common in reliability studies investigating how the expected life time of a product changes with some operational stress variable X. This model assumes that the life time, Y, has an exponential distribution whose parameter λ depends on the value x of X.

We write λ(x) to indicate the dependence of the parameter

λ on the value of the stress variable X. An example of such a regression model is log λ(x) = α + βx. Suppose that in a reliability study, the stress variable X is uniformly distributed in the interval (2, 6), and the above exponential regression model holds with α = 4.2 and

β = 3.1.

(a) Find the expected life time of a randomly selected product. (Hint. Given X = x, the expected life time is 1/λ(x) = 1/ exp(α + βx). Use the Law of Total Expectation. Optionally, R may be used for the integration.)

(b) Give the joint PDF of (X,Y). (Hint. Use the principle of hierarchical modeling.)