Poisson Distribution from first arrival? Sorry if the question is too obvious or strange, I'm learning about Poisson distributions by myself. 
Say I have some independent process that follows a Poisson distribution of unknown rate (10 particles in 1D that have to get to a specific position where the process finishes) and that I know the mean time at which the first particle gets to that position (I run simulations and I can identify the first time a particle reaches that position, for example). How can I estimate the mean time at which the 10 particles have arrived to that position? 
Thanks a lot for any help! I thought it was just 10*time of the first one but that doesn't seem right, does it?

Let the parameter of the Poisson be $\lambda$. Then the distribution of the minimum arrival time is exponential with mean $\frac{1}{10\lambda}$. If we know that then we know $\lambda$.
To finish, we use the moderately standard result that the mean of the maximum of $n$ independent exponentially distributed random variables with parameter $\lambda$ is 
$$\frac{1}{\lambda}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right).$$
