Does the expected value of the residual time (the amount of time one has to wait) equal the expected value of the spent time (the amount of time since the last arrival)? I know that the expected value of the residual time for a general distribution is $\frac{E[X^2]}{2E[X]}$, where $X$ is a r.v. denoting the time between events. Unsure for the spent time though.
2026-03-12 22:20:06.1773354006
residual waiting time and spent time
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Answered in comments; yes, their distributions are the same. This article talks about it in more detail for anyone interested.