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We show analytically that the answer to the question, "The longer it has beensince the last earthquake, the longer the expected time till the next ?"depends crucially on the statistics of the fluctuations in the interval timesbetween earthquakes. The periodic, uniform, semigaussian, Rayleigh andtruncated statistical distributions of interval times, as well as the Weibulldistributions with exponent greater than 1, all have decreasing expected timeto the next earthquake with increasing time since the last one, for long timessince the last earthquake; the log-normal and power-law distributions and theWeibull distributions with exponents smaller than 1, have increasing times tothe next earthquake as the elapsed time since the last increases, for longelapsed times. There is an identifiable crossover between these models, whichis gauged by the rate of fall-off of the long-term tail of the distribution incomparison with an exponential fall-off. The response to the question for shortelapsed times is also evaluated. The lognormal and power-law distributions giveone response for short elapsed times and the opposite for long elapsed times.Even the sampling of a finite number of intervals from the Poisson distributionwill lead to an increasing estimate of time to the next earthquake forincreasing elapsed time since the last one.

The paradox of the expected time until the next earthquake