I'm trying to do what I thought was a basic problem but doesn't seem to be working out properly. I'm looking at disproving a claim that tides cause earthquakes, where the main mechanism that people claim is that it happens during a new or full moon, and especially during a perigee moon (when it's closest to Earth).
People like to give ±1 week windows which is basically half a lunar cycle, so it seems basic that the probability an earthquake would happen, given random chance, would be 50% that it would happen within a week of, e.g., a full moon. Similarly, since perigee/apogee happens roughly on the same timescale, it seems as though it would be 50% chance that an earthquake would happen within 1 week of a perigee moon. Put the two together and you have a 25% chance that an earthquake would happen within both a perigee and full moon, correct?
(Important note, updated: Actual period between new/full moons averages 29.52 days, actual time between perigee moons averages 27.56 days. However, apogee and perigee are NOT normally distributed. Apogee happens a mode of 27.78 days (mean is 27.55±0.27), while perigee is much more asymmmetric, having a mode of 28.4 days at the peak of approximately a Lorentzian. Half the max of the Lorentzian is ±0.08 days. But, the mean is 27.56 with a standard deviation of 1.12; the range is 24.6-28.6 days. I'm thinking this could throw off the modeling?)
Assuming that's correct, I seem to be running into issues when trying to figure out the probability that an earthquake would happen by chance within ±X days of BOTH a perigee and full moon. I thought the equation would be simply (2*X/(# days in lunar month, 29.5))*(2*X/(# days between perigee, 27.5)).
However, when I do a monte carlo simulation with 500,000 randomly chosen dates within the time period of 1933 to 2012 (just happens to be when I have earthquake data), the fractions do not line up. For example, the simulation shows that 14.4% should be within 5 days of both a full and perigee moon, but my above math says it's only 12.3%.
I have checked the results of my simulation against days from perigee, apogee, new moon, and full moon times. As expected from a random distribution with a large $N$, the number of times that the simulated earthquake is a given time period away from maximum perigee/apogee/new/full moon is even. Except for perigee, where I see a fall-off for $>|±12.5|$ days from when it's closest to perigee. I'm thinking this has to do with the non-Gaussian distribution of perigee times? And could that account for the 2% difference at the 5-day example?
Is the best way to approach this, because these perigee times are a bit crazy, to simply go with the Monte Carlo results?
P.S. This has been updated to better reflect a correction I made in my data. When I initially posted this, I had some incorrect full/new moon dates in my table that were throwing some results off.