Hi,

the number of satellites launched before is Poisson with parameter and, given , the times of the launches are (without order) i.i.d. random variables uniformly distributed on , ok?

Let be their lifespan: i.i.d. with distribution .

If you prove that , then you see that the number of satellites that are still working at time is obtained by picking each of the launched satellites with this same probability, hence is Poisson with parameter . (It's the usual "picking each point of a rate Poisson process with probability yields a rate Poisson process"). You can also check this conclusion by computing explicitly: , etc. I let you fill in the dots... If you need further help, please tell us extendedly what you tried and so on.