The length of time that a machine operates without failure is denoted by X and the length of time to repair a failure is denoted by Y.
X and Y are independent each with exponential distribution, mean 1.
a.) Find the pdf of U = X/[X + Y], the proportion of time that the machine is in operation during any one operation-repair cycle.
b.) Use the pdf of U to find the probability that U is greater than 0.75.
For part a, I did:
f(x) = λ*e^(-λx) for 0 < x
F(u) = P( U < u ) = P( x/(x+y) < u) = P ( x < u/(1-u) ) = integral from 0 to u/(1-y) of λ*e^(-λx) dx
I'm wondering if I set that up correctly?