The Always Ready Battery Co. has developed a new "Failsafe" battery which incorporates a small secondary battery that becomes immediately functional upon failure of the main battery. The operating life of the main battery is a gamma distributed random variable as \(X_{1} \sim\) Gamma(3,1) where \(X_{1}\) is measured in years. The operating life of the secondary battery is also gamma distributed as \(X_{2} \sim \operatorname{Gamma}(2,1)\). The operating lives of the main and secondary batteries are independent.

(a) Let \(Y_{1}=X_{1}+X_{2}\) represent the total operating life of the Failsafe battery, and \(Y_{2}=X_{1} /\left(X_{1}+X_{2}ight)\) represent the proportion of total operating life that is contributed by the main battery. Derive the joint probability distribution of \(\left(Y_{1}, Y_{2}ight)\).

(b) Are \(Y_{1}\) and \(Y_{2}\) independent random variables?

(c) Define the marginal densities of \(Y_{1}\) and \(Y_{2}\). What specific families of densities do these marginal densities belong to?

(d) What is the expected proportion of total operating life contributed by the main battery? What is the probability that the secondary battery contributes more than 50 percent of the total operating life of the Failsafe battery?

(e) What is the expected total operating life of the Failsafe battery?