Expected time of the first of six downloads

Question

Downloads of an app occur as a Poisson process over a two-hour window, [0, 120] minutes. Exactly 6 downloads happened. Given this, what is the expected time (in minutes) of the first download?

Accepted Answer

Conditioned on 6 downloads, the six times act like 6 independent uniform points on [0, 120], sorted. The first download is the smallest of the six. For n uniform points on an interval of length T, the sorted points cut the interval into n + 1 equal-on-average gaps, so the k-th sits at 	frac{k}{n+1}T. The first is k = 1: E[	ext{first download}] = \frac{1}{6 + 1}	imes 120 = \frac{120}{7} \approx 17.1 	ext{ minutes}. So the first of the six is expected about 17 minutes in. With more arrivals crammed into the window, the first one is pushed earlier, exactly what 	frac{1}{n+1} captures. Takea…

Expected time of the first of six downloads

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