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?
Yves is correct. Uniform distribution appears to be the correct answer when you condition on the interval.