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Probability that a 6.0 player can join Mochy's 4.0 club

Posted By: Timothy Chow
Date: Wednesday, 9 April 2014, at 12:26 a.m.

In Response To: Probability that a 6.0 player can join Mochy's 4.0 club (Jason Lee)

Jason wrote:

What is the justification for the errors being exponentially distributed? Do you just have empirical evidence, or is there some intrinsic nature of errors that cause them to be distributed as such?

Is k the number of errors in a match? If so, I can tell you, as somebody who approximates a typical 6.0 player, I'm shanking way more than 4 moves in a match. Way more.

Bob Koca asked a similar question. You might find the heuristic justification more convincing if you interpret k as the number of games in a match and the exponential distribution as modeling the errors in a game. However, it isn't my intention to interpret the parameters that literally. What I was looking for was some way to model a distribution that arises from adding together a bunch of positive-valued random variables, each of which takes on larger values with decreasing probability. The gamma distribution has that property.

You could of course come up with a more complicated model where you can give more convincing theoretical justifications of each individual component, as you suggest: Start with some distribution that you can theoretically justify as modeling an individual error, and add together k of them, where k itself follows some kind of probability distribution that you can theoretically justify as modeling the arrival rate of errors. My prediction is that you'll end up with something that looks roughly like a gamma distribution anyway: the distribution will be zero for negative values, and will be unimodal and positively skewed. The process of adding up independent random variables tends to wash out the details of the component distributions. Given that, and given that we're just trying to get some feeling for the probability in the subject line, I decided to pick the gamma distribution for simplicity.

Note that about a month ago, David Presser suggested using a chi distribution instead. One probably has to wave his hands even more furiously to justify that choice, but I'd guess that carrying out the same calculation I did with the chi distribution instead of the gamma distribution would give a similar answer in the end. I picked the gamma distribution over the chi distribution only because I wanted to pick something that directly modeled summing a bunch of random variables.

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