#### Posts

2019-05-16: Glow worms return

2019-04-11: Original memetic sin

2019-01-31: The theory of weight

2018-11-06: Origins of telephone network theory

2018-10-24: Modern thought

2018-09-10: Feeding a controversy

2018-06-11: Glow worm distribution

2018-04-23: Outlawing risk

2017-08-22: A rebuttal on the beauty in applying math

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: MathEpi citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-20: Random dispersal speeds invasions

2014-04-14: More on fairer markets

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

## More stochastic than?

In some current research, I need a way to compare distributions and talk about one probability measure that is more stochastic than another. After a little effort (wheel re-invention and directed literature search), and I happened on the very cool idea of 2nd-order stochastic dominance. But here is a different approach, just for kicks.

The basic idea is that given two measures with the same expected value, measures that exhibit more variation around the center are dominated by measures that exhibit less. So singleton $\delta$-functions are dominates over all other measures.

Let's say that for two measures $f$ and $g$ centered at $0$, measure $f$ is dominated by measure $g$ if and only if

$\forall x, \quad \int_{0}^{x} \int_{0}^{v} f(u) d u d v \geq \int_{0}^{x} \int_{0}^{v} g(u) d u d v$

Now, just looking at this doesn't make much sense to me -- no intuition. So to get a better handle, it's useful to actually have an example we can study. One of the easiest measures where we can apply this is the logistic measure. If the probability distribution is

$p(x;a)=\frac{1}{a \left(e^{x/2a} + e^{-x/2a}\right)^2}$

then the lowered CDF is

$P(x; a) := \int_0^{x} p(x,a) dx = \frac{e^{x/2a} - e^{-x/2a}}{2 (e^{x/2a} + e^{-x/2a})}$

and the integrated CDF which we need for comparison is

$I(x; a) := \int_{0}^{x} P(x; a) d x = a \ln\left(\frac{e^{-x/2a} + e^{x/2a}}{2} \right)$

We can plot a few examples (below). And what we see is that as $a$ increases and the measure gains more variation, the CDF integrated from the mean is lower because there is less probability near the mean. The delta-function which integrates to an absolute value dominates everything. This is provoking, in that it relates scalar probability distributions to a subset of convex functions, and convex functions generally have very useful properties when we consider optimization problems.