13 November 2007

Two examples of convolution

1. This morning it was raining in Philadelphia. I walked to school, and as I was walking I considered the question -- which route should I walk so that I get the least wet? (I have an umbrella, but it is a rather small one because I don't like carrying a full-sized umbrella, so I still get a bit wet.) There is one particular street I could have taken that has a lot of trees. I find that when I walk on this street while it's raining I get rained on less, because the trees keep some of the water from falling to the ground immediately. But all the rain eventually gets down to the ground (I'm assuming; perhaps some of it gets absorbed by the leaves?) and roughly speaking there is some sort of convolution here, where the "rainfall density function" gets convolved with some sort of operator that shifts the rain forward in time. The net effect is that sometimes you can be walking down a tree-lined street after the rain stops and yet it still falls on you.

2. Last night, grading exams. Conventionally at Penn in the calculus courses we set the curve on the exams so that 30 percent of the grades are A, 30 percent are B, 30 percent are C, and 10 percent are D or F. This seems to a lot of people like too many A's and C's and not enough B's. But it makes sense; since it will not be the same students that get the A's on every exam, my advisor (who also happens to be the coordinator for the course) said that the distributions for the various exams end up effectively getting convolved to give the final distribution of grades, which tends to end up more like 25-40-25 than 30-30-30. (This isn't entirely true, though, as the exam grades are correlated with each other; students who do well on one exam tend to do well on the others.)