Gaussian Function in Statistics

[QuarkCharmer]

Homework Statement

Not really a homework problem, but something that I was curious about. I was thinking about how a calculator finds the area under the standard normal distribution, and I started to assume that it most likely has the antiderivative for the function that makes the standard normal curve, and it simply substitutes the various z-score values for the F(b)-F(a) part of the fundamental theorem of calculus.

Homework Equations

The Gaussian Function:
$$f(x)=\frac{1}{\sqrt{2 \pi \sigma^{2}}}e^{\frac{(x-\mu)^{2}}{2 \sigma^{2}}}$$

The Attempt at a Solution

I know at my current math level, I have no hope of integrating this function, (thankfully wolfram and others can aid me), but I was wondering what they use for the mean(mu) and the variance(sigma), and how they came up with that? I think it would be handy to know this.

Alternatively, is the calculator using the Gaussian Integral to perform this task instead?
$$f(x)=e^{-x^{2}}$$

 

[Ray Vickson]

It is a rigorously-proven *theorem* that there is NO finite formula for the antiderivative of the Gaussian that involve only elementary functions (trig functions, exponentials, powers and inverses of all these). There are various infinite series, etc., but these are not "finite". Typically, numerical methods would involve approximate formulas of varying complexity and precision. The best of these might achieve full machine accuracy, but if expanded out to many more decimal places would start to reveal discrepancies between exact values (computed, say, using numerical integration) and the results from the formula.

RGV