This question arises while I am learning Continuous Time Markov Chain :

A machine is working for an exponential time with rate $\mu$ before breaking down. The repair time of the machine is exponentially distributed with rate $\lambda$. Find the probability that the machine is working at a certain time $T$.

Note : The pdf of an exponentially distributed random variable with rate $\lambda$ is $\lambda e^{-\lambda x}$

My idea is to conditioning on what time the machine starts breaking down and all I get is $$ P[\text{Machine working at time $T$}] = \int_0^T P[\text{Machine working at time T} | \text{Machine is breaking down at $x$}] \mu e^{-\mu x} ~ dx $$ and I couldn't proceed any further.

Any ideas on how to solve it? Thanks!


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