What is the closed-form expression for "cumulative density of a zero-mean unit-variance Gaussian" and one other? 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 
 A: (1)
The phi function $\Phi(x)$ you saw in the paper is defined as the integral
$$\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{t^2}{2}} \mathrm dt$$
where the function
$$\phi(x)=\frac1{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$$
is what they were referring to as the "zero-mean unit-variance Gaussian". One might alternatively see $\Phi(x)$ being expressed in terms of the error function $\mathrm{erf}(x)$:
$$\Phi(x)=\frac12\left(1+\mathrm{erf}\left(\frac{x}{\sqrt2}\right)\right)$$
(in short, what they called "cumulative density" (more properly, "cumulative distribution", as Michael points out.) here means taking the integral of the "probability density"; that is, you are integrating your Gaussian function.)
Any good statistics books should have a discussion of the Gaussian/normal distribution.
For methods on evaluating the error function on a computer, see this or this.

(2)
$p_i \sim \mathcal N(p_i;s_i,\beta^2)$ was in fact already explained in the sentence that brought it up: that is, the performance $p_i$ follows a Gaussian/normal distribution around the skills $s_i$, with $\beta^2$ being the variance.
