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# What is Return Period?

Your recent vocabulary may include “100-year event” (happening more often), (drainage system designed for) “10-year storm,” and so on, courtesy mainstream media and news outlets. For instance, in a recent article in Chicago Tribune, the author described the Brays Bayous of Houston and its design capacity.

Houston drainage grid ‘so obsolete it’s just unbelievable’

What exactly is the concept of return period? Does a 10-year return period event occur diligently every ten years? Can a 100-year event occur three times in a row?

If we define T as a random variable that measures the time between the events (wait time or time to the next event or time to the first event since the previous event), the return period of the event is the expected value of T, i.e., E[T], its average measured over a large number of such occurrences.

A 10-year return period event does not happen cyclically every ten years. If we average the wait times of a lot of such threshold exeedence events, we will get approximately ten years. Just like when you wait for the bus, you wait for short time or a long time, but you think of the average time you wait for a bus everyday, you can see events happening in a cluster or spaced out, but all average to an n–year return period.

Return period is the expected value of a Geometric distribution. Imagine we have a series of independent Bernoulli trials of 0s and 1s. 0 if the event does not happen (No Bob), 1 if the event happens (Yes Bob). The 1s can occur with some probability of occurrence p. Suppose, we have eights 1s (8 Bobs) in 79 years the probability of occurrence p = 8/79 = 0.101.

The expected value of the wait time (time between 1s) that is Geometrically distributed is the inverse of the probability of the event. Since the probability of Bob is 0.101, the return period (expected value of the wait times) is 1/0.101 ~ ten years. A 10-year return period event.

In lesson 34 of the data analysis classroom, we learn about return period through Bob and his reappearance. Bob’s time of occurrence also relates to Geometric distribution.

Thank you.

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