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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.

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    $\begingroup$ 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. $\endgroup$
    – Yves
    Commented Feb 16, 2023 at 10:14
  • $\begingroup$ 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. $\endgroup$
    – whuber
    Commented Feb 16, 2023 at 16:59

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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 $1/\lambda$).

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  • $\begingroup$ I suppose the question here is if the fact that only one event occurs means we should condition on that event. If the number of events wasn't specified then, your answer is certainly correct. $\endgroup$
    – Nic
    Commented Feb 16, 2023 at 16:40

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