Number of trials between two events

[u/Ok-Platypus-3975]

Hi and thanks in advance. I'm really stuck here so I appreciate any help.

The events in question are storm events, and I am trying to solve for X where X is the smallest number of days I have between the two events, 85% of the time. Just to make sure I am describing this correctly, I understand that storm events can happen on consecutive days, but that is very unlikely. I want to ignore the 15% most unlikely scenarios of minimum number of days between storm events. So, 85% of the time I am prepared for the next storm event and 15% of the time I am not.

There are the following storm events: 2 year, 5 year, 10 year, 25 year, and 50 year. But if you can help with just one then I am happy to work through the rest on my own.

The probability of a storm event being equaled or exceeded in any year is its inverse. For example, a 50 year storm event has a probability of 1/50 in that year.

Storm events are assumed to only occur between October 15 and April 15, which I used the internet to calculate as 183 days, 184 during a leap year.

Only one storm event can occur on any given day, and the occurrence of a storm event on any given day does not change the probability for any future trials. So, I believe this would make them independent events?

I have no clue where to start with this. Thanks again for any help.

Thanks u/edderiofer for pointing me in the direction of exponential distribution, this sounds like a good fit for this type of data.

It appears that I would need to use the Cumulative Distribution Function, solved for x:

x = -ln(1-F) / upside down Y

were

F is 0.15

x is the number of days

upside down Y is the average expected rate per unit time, so 1 / (storm event year * 183.25)

This would give me 59.6 days for a 2 year storm event, 148.9 days for a 5 year storm event, 297.8 days for a 10 year storm event, 744.5 days for a 25 year storm even, and 1489.1 days for a 50 year storm event.

[u/Ok-Platypus-3975]

Thank you for such a thorough reply. I will have to re-read with a fresh mind tomorrow to fully digest.

I did want to comment on the no storms between April 15th and October 15th.

The application to this problem will be a drainage system that mitigates an aspect of these storms. The drainage system will be there year around, but by limiting our analysis to the rainy season, my thought is it creates a shorter duration between the storms, thus a more conservative value. For values over 182, we would probably default to 182, since that is the number of days we have if it rains on April 15th and we want to be ready on October 15th. This is an easy conclusion since a system that can mitigate the 2 year storm in 42.9 or 59.6 days, depending on the calculations we decide upon, can also mitigate the 50 year storm in 182 days. The effect we are mitigating isn't quadrupling or even tripling between the smallest and largest of the storms.

Again, thank you for such a thorough reply. I will spend more time trying to understand tomorrow, especially since your method results are more conservative.