Poisson Process: Probability distribution to describe time (distance) to successful event? Given some length of time $t$ with successful events occurring in this interval at rate $\lambda$. Assume that only one successful event occurs during this interval of length $t$. Which distribution describes this location in the interval? This is definitely some kind of exponential distribution, but with what parameter?
edit: Does it make sense to condition on the fact that only one event happened in this interval? And use Bayes theorem to get the distribution?
second edit:
Yves is correct. Uniform distribution appears to be the correct answer when you condition on the interval.

I don't fully understand your question, but: one way of defining the Poisson Process is having a series of events with inter-arrival times defined by an exponential distribution with rate parameter $\lambda$ (or alternatively, expectation $1/\lambda$).
