probability question of 200 level course

Question

Ninety identical electrical circuits are monitored at an extreme temperature to see how long they last before failing. The 50th failure occurs after 263 minutes. If the failure times are modeled with an exponential distribution, when would you predict that the 80th failure will occur? At what time will only 5% of the circuits fail?

Answer 1

If 50 of 90 fail at 263 minutes, our estimate is that F(263) = 50/90

Then, F(t) = 1 - e^-λt ;1 - e^-λ(263) = 50/90; e^-λ(263) = 40/90

-λ(263) = ln(4/9); λ = ln(9/4)/263

Then, for the 80th failure, λt = -ln(10/90) = ln(9)

t = ln(9)/(ln(9/4)/263) = 263 * 2 ln(3)/(2 ln(3) - 2 ln(2)) = 263 ln(3)/(ln(3)-ln(2) = 263/(1 - ln(2)/ln(3)) = 712.6

If 5% fail, 95% remain. t = 263 ln(20/19)/ln(9/4) = 16.64