The return period concept

In a paper with David Sibaï (that can be found here), we had a discussion about the concept of “return period” in hydrology. Actually, it looks like this concept has been introduced by Emil Gumbel in his book on Statistics of Extremes.

Graphs are also proposed,

The link between a probability and time can be established clearly using the geometric distribution. The time of the first success has the following distribution

\Pr(X = k) = (1 - p)^{k-1}\,p\,

where 0< p \leq 1 denotes the success probability. Then 
\mathrm{E}(X) = \frac{1}{p},
\qquad\mathrm{var}(X) = \frac{1-p}{p^2}.

This means that the time we have to wait – on average – before the first success is then simply the inverse of the success probability.
Note that this distribution is simply the discrete version of the exponential distribution, satisfying the memoryless property (in the context of continuous variates). And actually, this geometric distribution is the only discrete distribution satisfying such a property.
For instance, to illustrate this idea, the 1910 flood in Paris was suppose to be a centenial event. It does not mean that we must have a similar event next year, it means that we have to wait (still) 100 year – on average – before having a similar event. Or similarly, such an event occurs every year with a 1% probability, assuming temporal independence  (this was actually the point we discussed in our paper).

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