"I have two batteries of different brands. Each has a lifetime defined by an independent random variable. The PDF of each follows
Brand A : fX(x) = { e^-x, if x>=0 and 0 otherwise
Brand B : fX(x) = { 3e^(-3x), if x>=0 and 0 otherwise
Both battery brands are tested side by side, starting at t = 0. As each battery runs out, it is instantly replaced.
1) What is the expected number of brand B failures until a given time t?
2) What is the PDF of the amount of time before the first failure of either battery brand?
3) What is the expected value of the time until the third Brand B battery fails? What is the variance of that time?
4) Given that a Brand A battery just failed, what is the expected value of the time we wait until the next Brand B failure?"