# How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a machine repair time exponentially distributed with respectively rate parameters $\mu_A$ and $\mu_B$. I would like to write the machine repair time as one cumulative distribution function G(s).

I know the probability of machine A breaking down first is the minimum of two exponentially distributed variables with parameters $\lambda_A$ and $\lambda_B$, which gives another exponentially distributed variable with parameter $\lambda_A+\lambda_B$:

$\frac{\lambda_A}{\lambda_A+\lambda_B}$, therefore the probability of machine B breaking down first: $\frac{\lambda_B}{\lambda_A+\lambda_B}$.

Can i now say that the $G(s) = \frac{\lambda_A}{\lambda_A+\lambda_B}(1-\exp(-\mu_At))+ \frac{\lambda_B}{\lambda_A+\lambda_B}(1-\exp(-\mu_Bt))$

*Your solution should have $s$ on the right hand side instead of $t$.