CDF*[1-CDF]/PDF --- name? integrable? Suppose I have a random variable $X\in\mathbb R$ distributed according to a smooth nonzero probability density function (PDF) $f(x)$, with cumulative distribution function (CDF) $F(x):=\int_{\infty}^x f(\bar x)\,d\bar x$.  The following quantity is showing up in a calculation I'm working on:
$$q(x):=\frac{F(x)\cdot[1-F(x)]}{f(x).}$$
Does this quantity have a name?  And, what are the conditions under which $q(x)$ is integrable?
For what it's worth, it appears to be the product of the Mills ratio and the CDF.
 A: Logistic curve
One relationship might be with logistic growth which is based on the following differential equation:
$$f'(x) = f(x)(1-f(x))$$
But then for $F(x)$ and inhomogeneous (using some variable rate $g(x)$)
$$F'(x) = g(x) F(x)(1-F(x))$$
So if we express the CDF as a logistic curve
$$F(u) = \frac{1}{1+e^{-u}}$$
where the parameter $u$ is an integral of $q(x)^{-1}$ (where $m$ is the median for which $F(m) =0.5$)
$$F(x) = \frac{1}{1+e^{-\int_{m}^x q(t)^{-1} dt}}$$
Then
$$f(x) = F'(x) = F(x)(1-F(x)) q(x)^{-1}$$
or like your expression
$$q(x) = \frac{F(x)(1-F(x))}{f(x)}$$
A related relationship is that the log odds (odds based on the CDF) are
$$\log\left(\frac{F(x)}{1-F(x)}\right) = \int_{m}^x q(t)^{-1} dt$$
And $q(x)$ is the inverse of the rate at which the log odds increase.
Order distribution
The terms like $F(x)\cdot(1-F(x))$ also occur in the distribution of order statistics.
But I am a bit puzzled how you can get this $f(x)$ in the denominator. There are not so many expression where you use $1/f(x)$.
