The following is a silly little problem a friend and me got to talk about, but it got stuck in my mind and cannot shake it. A few days ago we caught in a heavy rain (lot of them these days) and, while running for cover, the rain was so heavy and dense that in the time it took us to get to a covered area, we were all wet, plus the rain was starting to "cool down". Partly also due to the previously sampled local flora (bevereggia familiaris), thoughts ensued:
Him: We might as well have stayed there, after all, if we consider the rain to be of uniform distribution, we would have spent the same time in the rain.
Me: Not quite: when we ran for cover we were moving, so we were actually continuously summing the areas. This means convolution, and the distribution converges towards a Gaussian, so our sides will be less wet.
Him: Yes, but the area would be the same. If we approximate ourselves with some basic area shape, that would hold the same amount of water in both cases.
Me: But the Gaussian bell and the rectangle (as viewed in a frequency domain) would only intersect until the Gaussian's $\sigma$, which would mean that the "skirts" of the bell would have fallen outside, thus less wet.
Other (counter-)arguments involved saying that, if the rain was continuous, that would have meant the distribution would have also changed into a Gaussian for the whole area of rain, and that the analogy of the square and the Gaussian distribution, in the frequency-domain, can be scaled down to fit us, so the skirts are not lost, but then the rectangle is no longer representing us since it's scaled down to fit the bell, but then...
The discussion ended up in a stale-mate, all the more so since we were already soaking wet and the local flora must have contained venomous bugs that altered our clarity of thought, but the answer won't quit me. Since neither my friend, or I, are mathematicians (electronics, both), could someone please let me know if the reasoning was sound, and if yes, what would be a correct answer? Would the continuous rain have a uniform, or other distribution? Would a static vs. a moving object have different distributions?
For simplicity, consider a large square as the area of "action", with a uniform distribution of noise over a period of time, where there are two smaller squares: one static and another moving. Other simplifications can also be made, such as the two smaller squares being in the center, next to each other, and the moving one going linearly to the side for the whole duration of the "noise", for example.