From: Osher Doctorow on
From Osher Doctorow

It turns out that the Logistic probability distribution is also fat-
tailed or heavy-tailed, which is important for Probable Causation/
Influence (PI) since the Riccati Differential Equation and its
Logistic Differential Equation and Exponential Growth/Decay subtypes
are important in PI.

See Wikipedia's online "Kurtosis", "Logistic Distribution", "t
Distribution" or "Student's t distribution", etc. The Gaussian/
Normal distribution has 0 excess kurtosis (a measure of fat-tailed and
peakedness at least for symmetric distributions), while the Logistic
Distribution has excess kurtosis 6/5 and the t-distribution has excess
kurtosis 6/(k - 4) for k > 4.

It is difficult to compare these with the exponential distribution
because the latter is nonnegative asymmetric while the others are both
negative and positive on the whole real line and are symmetric, and
the excess kurtosis of the exponential distribution is enormously
distorted compared to the others because of this (it is 6).

The logistic probability distribution is just the logistic
differential equation except that the derivative is replaced by the
defining symbol of the logistic probability density function in the
former.

Osher Doctorow
From: Osher Doctorow on
From Osher Doctorow

Recall that the logistic differential equation:

1) dy/dt = ky(1 - y) for y normalized into [0, 1] (that is, y a
variable in [0, 1]), k real constant (often taken > 0)

is an expansion-contraction equation in the sense that y (for example,
population percent of a particular group or category) tends to grow or
expand/increase, while 1 - y represents its tendency to contract or
decrease due to such things as environmental constraints (inability of
the environment to support a population as it gets too larger,
etc.). It can be solved by separation of variables and partial
fraction decomposition of y(1 - y) and turns out to be a ratio of
(linear functions) of exponentials or negative exponentials in time.

Osher Doctorow