Absolutely IV

 

With A Little Help From My Friends

Would you believe in a love at first sight?
I'm certain it happens all the time, yeah
What do you see when you turn out the light?
I can't tell you, but it sure feels like mine.

How certain are you that it happens all the time?

A random equation has two parameters: 1) its location, mean/median/mode, and 2) its scale, variance, uncertainty. The adage is that nothing is certain except death and taxes. Given that you can cheat on taxes, but you can’t cheat death, I would suggest that only death is certain, and thus life is uncertain, i.e. has a variance.

An exponential distribution also has a variance but is defined only for positive numbers. This restriction is identical to saying that its location is zero. It still has a scale parameter, a variance, that is given as λ. It is suggested that the exponential distribution is a distribution of the absolute. It can be coordinate transformed by translation to any location, µ, as long as µ>0 and then its Probability Density Function, PDF, becomes

λ*e^-λ*(x-µ)

and its Cumulative Distribution Function, CDF, becomes

1-e^-λ*(x-µ)

The median of an exponential distribution is generally given as ln 2/λ, but this is when the location is zero. With the translation of the location, the median is ln 2/λ+µ. The mean is 1/λ+µ.

A normal logistics distribution has both a location and a scale parameter. Its CDF is

½*tanh((x-µ)/(2s))+½.

For the median of the two distributions to be equal requires that s = (ln 2)/λ and that µ>0.

When the location of a logistics distribution is zero, then its upper half, above its median, looks like an exponential distribution with a location of zero. This is hardly surprising. The exponential distribution is also the equation of radioactive decay. Its scale parameter, λ, is then known as a half-life.

There is no need as Grushka (Grushka, 1972) and Reyes (Reyes, Venegas, & Gómez, 2018) wo each have proposed to combine an exponential with a normal (e.g. Gaussian or logistics) distribution.  An exponential distribution is only the upper half of a normal logistic distribution with a location of zero. That does not mean that a logistics distribution is an absolute. An exponential distribution, an absolute, is half of a random normal logistics distribution, life.  Of this I'm certain.

References

Grushka, E. (1972). Characteristics of Exponentially Modifed Gaussian Peaks in Chromatograhy. Analytical Chemistry Vol 44, pp. 1733-1738.

Reyes, J., Venegas, O., & Gómez, H. W. (2018). Exponentially-modified logistic distribution with application to mining and nutrition data. Appl. Math 12.6, 1109-1116.