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Dirt in the Mouse & Glivenko-Cantelli Theorem

There was something wrong with the wheel of my mouse these days: it could not control the scrolling of a page very well. Today I found a screwdriver and opened the mouse. Then I knew the reason: there was a lot of dirt beneath the cover!

However, obviously the dirt would not follow a B(n, p = 0.5) distribution. The two proportions of the dirt in front and back of the wheel were not equal – there was much more dirt in the back. Why?

The Glivenko-Cantelli Theorem tells us that the empirical distribution converges to the true distribution almost surely.

So what? The proportions of the dirt has revealed my habit of using the mouse, as I have used it for too many times. What I usually do is to scroll down a page, and rarely will I scroll up – that is the real distribution, and the dirt in the mouse has recorded the empirical distribution day by day. If I am serious enough, I can get the numbers of the weight and estimate the parameter p in the Binomial distribution. (Perhaps Michal will again suggest me do this. 😁 )