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**Anonymous1** Starting at some fixed time, $\displaystyle 0,$ satellites are launched at times of a Poisson process with rate $\displaystyle \lambda.$ After an independent amount of time having distribution $\displaystyle F$ and mean $\displaystyle \mu$, the satellite stops working. Let $\displaystyle X(t)$ be the number of satellites working at time $\displaystyle t.$

(a) Find the distribution of $\displaystyle X(t).$

(b) Let $\displaystyle t \rightarrow \infty$ in (a) to show the limiting distribution is $\displaystyle Poisson(\lambda\mu).$