Cumulative distribution function: what does $t$ in $\int\exp(-t^2)dt$ stand for?

Question

I'm trying to teach myself how to quickly translate many different types of equations into VB, T-SQL and MDX code. Since I'm trying to build a skill, not just solve a single isolated problem, I'm try to figure this stuff out on my own as much as possible - but I'm stumped by the error function used in the calculation of the normal cumulative distribution function. In the equation I retrieved from the Wikipedia page "Normal Distribution," there is a $t$ symbol ($-t^2$, actually) which I can't find any references on.

$$\text{erf}(x)=\frac{1}{\sqrt{(\pi)}}\int_{-x}^x\exp(-t^2)dt$$

Can anyone tell me what it means, and how to derive that value if it's not obvious? I can't figure out if it's a common calculus symbol or a measure used in statistics. Could it be one of the parameters used in integral differentiation, as mentioned at http://en.wikipedia.org/wiki/Parameter and http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign ?

In addition, I've read that the CDF for a normal distribution is difficult to define precisely, so approximations based on things like Taylor Series and McLaurin Series are used. I want to figure out how to code such things on my own, if possible, but I was wondering if anyone had any thoughts on which method might be higher performing and which would yield greater precision.

Answer 1

$t$ is just a place-holder for a number: you're integrating the function $\mathrm{e} ^{-t^2}$ from $t=-x$ to $t=x$. The reason it's not defined where you read it is that to evaluate the error function $\operatorname{erf}(x)$ you need to provide a value for $x$, but not for $t$, which is a bound variable—it plays a purely internal role in the function. It's like $k$ in this formula for the sum of the first $n$ integers: $$\sum_{k=1}^{k=n} k=\frac{n(n+1)}{2}$$

PS Nothing to do with differentation under the integral sign, just plain integration.

PPS In the terminology of the Wikipedia article, $x$ is the "parameter on which the integration depends" & $t$ is the "parameter of integration" a.k.a. dummy variable (I've never heard of "parameter of integration" before").

Answer 2

As per $t$, it is indeed the "dummy variable of integration" -and we use it extensivelty in CDF's to avoid confusion with the limit of the integral, i.e. instead of writing

$$P(X\leq x) =\int_{-\infty}^x f(x)dx$$

we write

$$P(X\leq x) =\int_{-\infty}^x f(t)dt$$

since it is rather awkward to write something that verbally would translate into "$x$ ranges from minus infinity to $x$"

As for approximating the normal CDF, ch. 3 of Patel, J. K., & Read, C. B. (1996). Handbook of the normal distribution. has I believe relevant information.

Answer 3

Imagine someone came to you with some complicated 2D shape, and wanted to find the area of it.

You tell them "Oh, here's a way you can approximate it - put a grid of tiny squares over it and count how many squares are contained inside its boundary."

To keep things straight (so they don't count haphazardly), you then lead then through numbering the squares, row by row.

The $t$ is in effect playing the role of "which square are we up to?"

Clearly it plays no role whatever in the value of the area (since you could change the numbering order without changing the area nor your estimate of it). It's just a placeholder (a 'dummy variable'). The area isn't a function of $t$, it is just approximated by an approach that is using it as a way of keeping track of where you are in the calculation.