Suppose we have $n$ batteries which has a lifetime that is exponentially distributed with parameter $\lambda$. Each battery's lifetime is independent.
If we initially put 2 batteries on and every time a battery fails, we replace it with another battery until there is only one working battery.
What is the probability that we can use these $n$ more than $x$ years. And what is the expectation of the total time until there is only one battery left working.
I realize that we need to use the memoryless property of exponential distribution r.vs but I kind of got stuck there.
Best Answer
Hints:
The failure of the batteries is a Poisson process
Having two batteries on at the same time doubles the intensity of the process
You can use the $n$ batteries up to the point at which there have been $n-1$ failures