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I'm working on a hobby project that involves tournament-style player rankings, and I'm using the TrueSkill system developed by Microsoft for online gaming. It's a great system, but much of the math is far above my head. Even without understanding it, I've got most of what I need figured out, except for two equations which are never explicitly defined. The document that best outlines them is http://research.microsoft.com/pubs/67956/NIPS2006_0688.pdf.

(1) On the first page, equation 1 references psi, which denotes "the cumulative density of a zero-mean unit-variance Gaussian."

(2) In the preceding paragraph, there is mention of a function, N, representing player performance.

Having never taken a day of statistics in my life, and certainly not the graduate-level math that it seems is necessary for understanding this paper, it'd be great if somebody who does understand could just give me a numerical formula for these two functions that I could plug into my code (programming project). If you're curious, or it helps to answer the question, another great article about the system which is a little more approachable is http://research.microsoft.com/en-us/projects/trueskill/details.aspx.

Thanks for whatever help anyone

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    http://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function for $\Phi$. $\cal N$ does not give the player's performance, but rather the "probability distribution" of the performance. I'm fairly certain that for $\cal N$, use $f(p_i; s_i,\beta^2)$, where $F$ is as in the wiki link right above $\Phi$.2011-11-21
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    Whoever wrote the words "cumulative density" was at best not paying attention to what they wrote and not an expert on what they wrote. There is no such thing as "cumulative density". It's an oxymoron, and whoever wrote it is probably a moron. The word "cumulative" contradicts the word "density".2011-11-21

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