Satellites are launched into space at times distributed according to a Poisson process with rate $\lambda$. Each satellite independently spends a random time (having distribution $G$) in space before falling to the ground. Find the probability that none of the satellites in the air at time $t$ was launched before time $s$, where $s < t$.
I'm practicing for an exam and I am seeking a solution to this question. I have a solutions guide but the answer is presented without any explanation. I do not know how to quite arrive to an answer here as I am not very familiar with Poisson processes. Thank you.
Best Answer
If $G$ is the cumulative distribution function of the lifetime of the satellite, a satellite that was launched at time $x$ has probability $1 - G(t-x))$ of still being up at time $t > x$, independent of all other satellites. Thus the launches of satellites still up at time $t$ form an inhomogeneous Poisson process of rate $\lambda (1 - G(t-x))$. The number of these that were launched in the time interval $(a,s)$ is then a Poisson random variable with parameter $\mu = \lambda \int_a^s (1-G(t-x))\; dx$, and the probability that this is $0$ is $\exp(-\mu)$.