What is the name of the distribution with a probability density like $1/(1+\exp(x))$?

Please forgive my ignorance, what is the name of the distribution with a probability density like this? $$p(x) \propto \frac{1}{1 + e^x},\quad x > 0\,,$$ or more generally $$p(x) \propto \frac{1}{1 + \alpha e^{\beta x}},\quad x > 0\,,$$ or $$p(x) = \eta \frac{1}{1 + \alpha e^{\beta x}},\quad x > 0\,,$$ where $\eta$ is a normalising constant.

3 Answers

This is identical to a common distribution in physics called the Fermi-Dirac distribution, which describes a situation called Fermi-Dirac statistics. In a certain setting in physics, the average number of particles with an energy $\epsilon$ is $$\bar{n}_\epsilon = \frac{1}{e^{(\epsilon -\mu)/kT}+1}$$ where $\mu$, $k$, and $T$ are physical parameters that probably aren't so important to you (the chemical potential, Boltzmann's constant, and the temperature). Its trivial to reinterpret this as probability density function for the energy of a particle.

• thanks! since I also have an $\alpha$ parameter that I can adjust, do you since I should call it "extended / generalised / modified FD distribution"? or do you have any suggestion on this type of naming convention? – gwding Dec 15 '16 at 19:19
• @gwding Your alpha parameter corresponds to the "chemical potential" parameter in the F-D distribution: $\alpha = \exp(-\mu/kT)$. It might be the case that you gain more insight into the nature of the distribution by using something closer to the physics form, with $\exp(\beta(x-x_0))$ in the denominator. Coincidentally, $1/kT$ is often denoted $\beta$ in physics, so your naming convention for that parameter is the same! – jwimberley Dec 15 '16 at 19:26

The normalizing constant for the first should be $\frac{1}{\ln(2)}$ (not that it really matters for the present question).

I'm not aware of either having a name. The first (without the $\log(2)$ normalizing constant) is the survivor function for a truncated logistic distribution, but I haven't seen it used for a density function (though I expect that it has probably been named several times ... that's often the case with simple functional forms that are not in very wide use, where people "reinvent" such things without encountering previous ideas, which are often in different application areas*).

If you were to try to name it, then because of the logistic-type functional form you'd probably want to squeeze the word "logistic" in there somewhere, but the difficulty would be in choosing a name that would distinguish it sufficiently from the logistic density.

* and jwimberly's answer offers one such application area. The name "Fermi-Dirac distribution" seems a perfectly reasonable choice if you don't have a name in the application area you're working in.

A density that integrates to unity over $[0,\infty]$ would be

$$f_X(x) = \frac {\theta}{\ln 2}\frac{1}{1+e^{\theta x}},\;\;\; \theta >0$$

Raw moments are given by

$$E(X^k) = \frac {(1-2^{-k})}{\ln 2} \frac {1}{\theta^{k}}\cdot \Gamma(k+1) \cdot \zeta(k+1)$$

where $\Gamma()$ is the Gamma function and $\zeta()$ is the Riemann zeta function. So

$$E(X) = \frac {\pi^2}{12\cdot \ln 2}\theta^{-1} \approx 1.1866 \cdot \theta^{-1}$$

$$E(X^2) \approx \frac {7.212}{4\cdot \ln 2}\theta^{-2} \approx 2.601 \cdot \theta^{-2}$$

leading to

$$\text{Var}(X) \approx 1.193 \cdot \theta^{-2}$$

Numerical calculations verify these.

• How does this answers the question? – Jorge Leitao Dec 15 '16 at 10:51
• @J.C.Leitão I tend to take a broader view of "what the question is". See this post, meta.stats.stackexchange.com/q/2158/28746, where I advance my argument and offer also some cv-data to back it up. Also, offering the moment expression for a distribution that has not been studied, is useful kowledge for anyone interested in using the distribution. – Alecos Papadopoulos Dec 15 '16 at 11:54
• I agree with your conclusions in the meta you mentioned, "The moments of this PDF are X" is not a broader answer to the question "How is this PDF named?". The distribution has also been studied before, like other answers refer. – Jorge Leitao Dec 15 '16 at 13:54
• @J.C.Leitão As I already wrote, my main concern and criterion is whether it is relevant and useful information to be present in this thread- and it is. The connection to the Fermi-Dirac distribution has been noted in the other answers but I didn't see an explicit expression for the raw moments. It is a usual phenomenon for answers in CV to not be competitive but complementary, each providing some useful bit of information/knowledge. – Alecos Papadopoulos Dec 15 '16 at 14:14