I'm trying to identify a distribution presented to me as the "Van Loon distribution".

Question

My professor's book on Electrical Measurements presents 3 distributions that model the probability density function of the error an observation is expected to have given a set of already obtained observations.

These are the normal distribution, the "Van Loon" distribution and the Laplace distribution. The problem is that I can't find any other reference to a Van Loon distribution on Google and I suspect my professor is using a lesser known name for an otherwise common distribution.

The distribution is: $$\psi(\delta) = \frac{\alpha e^{\alpha \delta}}{(1+e^{\alpha \delta})^2}$$

Has anyone seen this distribution before?

Answer 1

This seems to be the logistic distribution with location $\mu=0$ and scale $s = 1/\alpha$. As noted by Michael Hardy in the comments, the PDF is symmetric [about $\mu$].

Answer 2

This is the logistic distribution. \begin{align} \int_0^\delta \frac {\alpha e^{\alpha\eta}}{(1+ e^{\alpha\eta})^2} \,d\eta = \int_2^{1+e^{\alpha\delta}} \frac{du}{u^2} = \frac 1 2 - \frac 1 {1+e^{\alpha\delta}} \to \frac 1 2 \text{ as } \delta\to+\infty. \end{align} Since the density is an even function, as may be checked with a bit of algebra, we have $$ p=\int_{-\infty}^\delta \psi(\eta)\,d\eta = 1 - \frac 1 {1+e^{\alpha\delta}} = \frac 1 {1 + e^{-\alpha\delta}} = \text{a logistic function of }\delta. $$ From this it follows that $$ \delta = \frac 1 \alpha \log \frac p {1-p} = \frac 1 \alpha \operatorname{logit} p. $$ "Logit" is conventionally pronounced with a "long o" as in "boat" and a soft "g" sounding like the "j" in "jet", and the stress on the first syllable.

Google the terms "logistic function" and "logistic distribution."