Stochastic process questions....
Does anyone know how to solve the below?
A tv controller needs 2 batteries to be operational. Suppose that, in addition to the tv controller, we have a set of 12 functioning batteries (battery 1, battery 2,.. and so forth.) Initially, we put in batteries 1 and 2 in the tv controller, leaving 10 spare batteries.
Whenever a battery (in the tv controller) fails, we immediately replace the failed battery by the lowest-numbered functioning battery that has not yet been put in use. Suppose that the batteries remain like new until they are installed in the tv controller. Suppose that the lifetimes of
the different batteries (in use in tv controller) are independent random variables, each with an exponential distribution having mean 4 months (independently of how the tv controller is used, even though that may be unrealistic).
T be the time that the tv controller ceases to work, i.e., the time that a working battery fails, causing the tv controller not to work, and Olga's stockpile of spares is empty. At that moment, exactly one of the 12 original batteries (which we will call battery N) will not yet have failed. (It will be the one working battery in the tv controller, even though the tv controller no longer works.)
(a) What is the expected value of T?
(b) What is the variance of T?
(c) What is P(N = 12)?
(d) What is P(N = 1)?
(e) What is the probability that exactly 5 batteries have failed during the first 8 months?