What is the specific name of this distribution?

I just can’t seem to find the name of this distribution:

$$\frac{e^{-x}}{(1+e^{-x}) ^2}.$$

From my understanding, it is generally applied to pandemics/epidemics.

None of the statistics books that I have looked at contained any information on it. I would really appreciate if someone can guide me about it.

• This is a density, not a distribution. See en.wikipedia.org/wiki/Logistic_distribution. Writing it as $p(1-p)$ with $p=1/(1+e^{-x})$ is very suggestive ;-).
– whuber
Commented Nov 28, 2023 at 20:53
• @whuher I would have taken the derivative of the logistic function as the cumulative distribution function to come up with this density. How would p(1-p) have any importance here beyond a "numerical accident". I don't see the connection Commented Nov 28, 2023 at 21:58
• @Ggjj11 The connections are many and deep. You might, for instance, recognize $p(1-p)$ as the variance of a Bernoulli$(p)$ variable as well as the Beta$(2,2)$ density. Moreover, the differential relationship $(1/p)^\prime=p(1-p)$ leads to many nice algebraic formulas for quantities associated with the logistic distribution. These all can be interpreted in terms of quantities meaningful in epidemics as well as more generally.
– whuber
Commented Nov 28, 2023 at 22:39
• Note that $\dfrac{e^{-x}}{(1+e^{-x}) ^2} = \dfrac{e^{x}}{(1+e^{x}) ^2}$ demonstrating the symmetry about $0$ of the standard logistic distribution Commented Nov 29, 2023 at 15:28

This is the PDF of a logistic distribution with its location parameter set to zero and scale parameter set to one.

In general, a logistic distribution has a PDF as follows, where $$\mu$$ is the mean and $$s$$ is the scale (related to, but not the same as, standard deviation and variance).

$$f_X(x) = \dfrac{ \exp\left( -\left(x - \mu\right)/s \right) }{ s\left( 1 + \exp\left( -\left(x - \mu\right)/s \right) \right)^2 }$$

If you set $$\mu = 0$$ and $$s = 1$$, you get the expression given in the OP.

The CDF is as follows.

$$F_X(x) = \dfrac{ 1 }{ 1 + \exp\left( -\left(x - \mu\right)/s\right) }$$

(Note that there is a technical difference between a PDF and a distribution, even if people often use the terms as synonyms in statistical slang. If you look at the edit history, I did it, too.)

• Tldr: start with the logistic function as cdf, then take the derivative to get the PDF in the form of the question Commented Nov 28, 2023 at 21:59
• Thank you so much for the help. Commented Nov 29, 2023 at 7:01