Monday, April 27, 2020

About Exponentials

The greatest shortcoming of the human race is our inability to understand the exponential function
- Albert A. Bartlett, The Essential Exponential! For the Future of Our Planet
Sounds profound; what does it mean to the PMO?

Consider communications:
the number of ways that N people (or systems or interfaces) can communicate is N*(N - 1), which for large N is approximately N-squared (an exponential)

Consider project finance:
The present value of future benefits of a project are discounted, exponentially, by the expected risk

Consider the so-called "bell curve" of natural clustering around the mean
The actual formula for the curve is complex, but it's core is the the 'natural number 'e'' raised to an exponential that involves the mean and standard deviation

Consider the decay of natural materials
This also is exponential

Consider the arrival rate of independent actors (events)
Again, an exponential, and an important concept in certain elements of risk management. 

It never ends!
Exponential increases, where the exponent is positive, may work to your advantage, but when the exponent is negative, the phenomenon is decreasing exponentially. Is this good for your project?
Perhaps, but, not so fast!
If the exponent is less than -1, there is a great flattening of the tail, to the point that the thing never ends! Yikes, will this ever be done? (*)

If you look up Bartlett's book, you'll find most of the chapters are available free in pdf format
Shout-out to herdingcats for the quotation

(*) Consider the natural number 'e' with exponents 0, 1, -1, -0.1, -0.001, respectively
The values, corresponding, are: 1.0, 2.7 ascending with positive exponent, then descending with negative exponents, respectively: 0.37, 0.9, 0.99, approaching but never reaching 1.0

Buy them at any online book retailer!

1 comment:

  1. An excerpt from my research paper (Barseghyan, P. (2020). Quantitative Models of Emergence in Light of Mathematical Theory of Human Systems - Part 3: Probabilistic Mathematical Models of New Quality Emergence; PM World Journal, Vol. IX, Issue V, May.)

    According to the central limit theorem, the distribution of the sum of identically distributed random variables with an increase in the number of terms asymptotically approaches the normal distribution if the variances of the terms are finite.
    Otherwise, when the variances of the terms are not finite, the distribution of their sum asymptotically approaches the so-called stable distributions.
    With regard to the duration of people's activities, this means that if for whatever reason people's actions begin to be delayed, their dispersion will increase accordingly, leading to the fact that the duration of the activity will no longer have a normal distribution, but with the increase of dispersion, the distribution will initially have right asymmetry and further increase in scattering will result in heavy tail distributions.