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Question

The time until the next request hits a server is exponentially distributed with a mean of 5 minutes. You have already been watching for 3 minutes with no request.

Given that nothing has arrived in the first 3 minutes, what is the probability that you must wait at least 5 more minutes?

Accepted Answer

The exponential distribution is "memoryless": the time already spent waiting has no effect on how much longer you must wait. Formally, for an exponential wait T, P(T s + t \mid T s) = P(T t). In plain words, the clock resets. Having survived the first 3 minutes tells you nothing new, so the extra wait behaves like a fresh exponential wait starting now. That means the 3 minutes already spent are irrelevant, and we simply need the chance a fresh wait exceeds 5 minutes. With mean 5 minutes, the rate is \lambda = 	frac{1}{5} per minute, so P(	ext{at least 5 more}) = e^{-\frac{1}{5}\cdot 5} = e^…

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