# 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.

• $E[q(X)]$ is related to the expectations of $X$ and the largest and smallest of an iid pair of such variables, and therefore exists if and only if $E[|X|]\lt \infty].$ (This holds whether or not $X$ has a continuous distribution.)
– whuber
Commented May 5, 2021 at 18:47
• For what it's worth, this eventually showed up in one of our research papers: arxiv.org/pdf/2012.06958.pdf Commented Mar 17, 2022 at 13:28

### 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

$$f(x) = F'(x) = F(x)(1-F(x)) q(x)^{-1}$$

$$q(x) = \frac{F(x)(1-F(x))}{f(x)}$$

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 $$1/f(x)$$.