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I have a Poisson process so its inter-event times are exponentially distributed.

Suppose I fix a time T, (=0, say), and ask what the distribution of the time of arrival of the first event after T is. How do I find this?

Could I reason as follows? Suppose the time of first event after T is t, and the time of the event previous to that is s. Then, s < T < t, and t-s is exponentially distributed.

Now p(t = a) = p(t-T = a-T) = p(t-s > a-T).

But I am worried that the last equality may not be true.

2 Answers 2