9.3.6 The Error Function

We have shown how to answer questions about normal distributions without obtaining a formula for the distribution function. Still, it would be nice to have such a formula. If we ask a typical computer algebra system for an antiderivative for `e^(-t^2text[/]2)`, we get the answer

Here "erf" is a function that the computer algebra system knows about and is prepared to evaluate, in the same sense that it is prepared to evaluate sine, tangent, exponential, and logarithm. When we ask the system to differentiate erf` text[(] t text[)]`, we obtain the derivative

Thus, whatever the function "erf" might be, it is an antiderivative of this function.

Checkpoint 9

The function erf is called the error function. Most computer algebra systems have some routine for calculating this function. It could be that such a system just approximates the integral in our last equation. We tested this conjecture by asking a typical computer algebra system to evaluate erf`text[(] 0.753 text[)]` and then to evaluate the integral

In addition to the answers, `0.713080` each way, this system gives the time of computation. The computation time for the integral was more than `8` times longer than the evaluation of erf`text[(] 0.753 text[)]`. Clearly, the system is not evaluating the error function by approximating integrals.

What is the computer system doing? For that matter, how does it find values of the sine, tangent, exponential, and logarithm functions? To answer these questions, we need to look into how we can use polynomials to approximate functions — a subject we will take up in the next chapter.