# 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.

• Given that the interval contains exactly one event, the distribution of the location (or time) of the event is uniform. This is a classical result. A good reference is the book by Sheldon Ross Introduction to Probability Models.
– Yves
Feb 16 at 10:14
• Welcome to CV, Nic. Would you please elaborate on what you mean by "develop"? Would that amount to estimating $\lambda,$ or do you intend some other interpretation, such as computing the distribution of the time of the event conditional on there being a single event (with $\lambda$ known)? BTW. basic relationships between Poisson processes and Exponential waiting times are developed at stats.stackexchange.com/questions/214421.
– whuber
Feb 16 at 16:59

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 $$1/\lambda$$).