The exponential's amnesia

Question

You model the time until the next trade on a quiet venue as exponential with rate $\lambda$ . You have already waited 10 minutes with no trade.

How much longer should you expect to wait, and what special property of the exponential does your answer illustrate?

Accepted Answer

The memoryless property. For an exponential waiting time T, the extra wait given you have already waited s has the same distribution as a fresh wait from scratch: P(T s + t \mid T s) = P(T t). Proof. Using the survival function P(T x) = e^{-\lambda x}, P(T s + t \mid T s) = \frac{P(T s + t)}{P(T s)} = \frac{e^{-\lambda(s+t)}}{e^{-\lambda s}} = e^{-\lambda t} = P(T t). The e^{-\lambda s} cancels, the 10 minutes already spent are completely forgotten. So the answer. After waiting 10 minutes, your expected remaining wait is still 1/\lambda, exactly what it was at the start. Waiting does not brin…

The exponential's amnesia

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