A transform function from $(−∞,∞)$ to $(0,1)$? I want to convert an integral from $(0, 1)$ range to $(-\infty, \infty)$ range by change of variable. What is the best transform function to do this - one that is simple, monotonic with $f(-\infty)=0$ and $f(\infty)=1$?
Thanks.
 A: One rather regular function is $\arctan : \mathbb R \to (-\pi/2, \pi/2)$, so you can take $f(x)=\frac{1}{\pi} \arctan x + \frac{1}{2}$.
A: $\newcommand{\logit}{\operatorname{logit}}$
$$
z=\logit p = \log \frac{p}{1-p}.
$$
As $p$ goes from $0$ to $1$, $z$ goes from $-\infty$ to $+\infty$ (provided the base of the logarithmic function is more than $1$; most often it is $e$.).  The logit of $1/2$ is $0$.  The graph is symmetric about $(1/2,0)$, so that, for example, $\operatorname{logit}\ 0.2= -\operatorname{logit}\ 0.8$.
This function is used in statistics.
The first syllable is pronounced with the "long o" sound as in "low"; the "g" like the "j" in "jet".
The inverse is the logistic function
$$
p = \frac{e^z}{1+e^z}\ = \frac{1}{1+e^{-z}}.
$$
If you're applying this to an integral, then which of many such functions should be chosen would depend on what integral it is.
Later edit: I occurs to me that if one wishes to be aware of the logit function, one show know this fact about probability:
$$
\logit \Pr(A\mid D) = \logit \Pr(A) + \log\frac{\Pr(D\mid A)}{\Pr(D\mid \text{not }A)}
$$
(and the letter $D$ may be take to stand for "data".)
