A space station requires the continual use of two systems whose lifetimes are independent exponentially distributed random variables X 1 and X 2 with respective rates λ 1 and λ 2 . When system 2...

A space station requires the continual use of two systems whose lifetimes are independent exponentially distributed random variables X₁
and X₂
with respective rates λ₁
and λ₂. When system 2 fails, it is replaced by a spare system whose lifetime
_X3
is exponentially distributed with rate λ₃, independent of the other systems. Find the distribution of the time Y = min{X₁, X₂
+ X₃} at which one of the systems becomes inoperative. Find the probability that system 1 will fail before system 2 (with its spare) fails.