Most likely time for an event that follows Poisson distribution Assume that I have a detector which is looking at cosmic events. These events follow a Poisson distribution with a rate $\lambda$ per minute.
What is the most likely time for the next cosmic event (assuming "now" is just a random moment)?
I understand how the probability of having no events after time $t$ has passed is:
\begin{equation*}
P(t) = \exp(-\lambda t)
\end{equation*}
But how can I move forward in calculating the probability distribution for the length of time between now and the next cosmic event?
I also found online that this exponential distribution is "probability distribution of the time between events in a Poisson point process" (Wikipedia). Would that mean that the most likely time is $t=0$? Does this make sense?
I appreciate any help in the right direction.
 A: The Poisson point process has the property that increments are independent, and for a homogeneous process, the interarrival times are exponentially distributed, therefore memoryless.
So if no event has been observed, the conditional distribution of the time until the next event remains exponential with rate $\lambda$.  That is to say, no matter how long you have observed the process without seeing an event, the mean time until the next event is constant.  For a process with intensity $\lambda$ events per minute, the mean (expected) time until the next event is $1/\lambda$ minutes.
However, you appear to be asking for the mode of the time until the next event ("what is the the most likely time").  In this case, the mode of the exponential distribution is $0$, because the interarrival time has density $$f_T(t) = \lambda e^{-\lambda t}, \quad t \ge 0$$ and the maximum of this function occurs at time $t = 0$.  This of course is not particularly useful, since it is not even a function of the intensity, meaning that the most probable time until the next event does not depend on how frequently events tend to occur.
A different way to conceptualize the time until the next event is to consider the median time, which is the time at which the probability of having observed the event will be $50\%$; i.e., $$\Pr[T \le t^*] = 0.5,$$ or $$t^* = \frac{\log 2}{\lambda}.$$
Ultimately, the most meaningful and informative aspect of interarrival times for a homogeneous Poisson process is the probability distribution itself, from which whatever statistics or probabilities you wish to calculate may be derived.
