Expected extra wait for the next quake (memoryless)

Question

The time until the next major earthquake on a fault is exponential with a mean of 10 years. Four quiet years have already passed with no quake.

Given the four quiet years, what is the expected additional wait until the next quake?

Accepted Answer

The exponential is memoryless: the chance of waiting another t years does not depend on how long you have already waited. Formally, P(T s + t \mid T s) = P(T t). The process does not "age" or build up pressure in a statistical sense. So the four quiet years give you no new information about timing. The expected additional wait is exactly the original mean: \mathbb{E}[T - 4 \mid T 4] = \mathbb{E}[T] = 10 	ext{ years}. This is unique to the exponential among continuous waiting-time distributions and is exactly why it models "completely random" arrivals. Takeaway. Because the exponential is mem…

Expected extra wait for the next quake (memoryless)

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