A machine is composed of two identical components placed in series. The lifetime of a component is a random variable having an exponential distribution with parameter μ. We have at our disposal a...

A machine is composed of two identical components placed in series. The lifetime of a component is a random variable having an exponential distribution with parameter μ. We have at our disposal a stock of n — 2 new components that we differentiate by numbering them from 3 to n (the components already installed bearing the numbers 1 and 2). When the machine fails, we immediately replace the component that caused the failure by the new component bearing the smallest number among those in stock. Let
T
be the total lifetime of the machine, and let
N
be the number of the only component that, at time T, will not be down. Find (a) the probability mass function of
N,
(b) the mathematical expectation of T, and (c) the distribution of
T,