The Standard Normal

Standard Normal curve is a special case of gaussian curve, which is kind of a inverse of e^k where is k=x^2. My curiosity was about how did this curve possess such important characteristic to quaalify for modeling most natural and fundamental processes such random distribution?

Turns out that we can slowly evolve this curver through other simple curves. We will start with polynomials first.

Here, h is a polynomial function with finite terms. Infinite sums lead to irrational zones. We have used only 20 terms here. Both second and third functions coincide. The third one is actually the guassian curve and second one is an approximation derived from the polynomial curve.

The following are further evolutions for Guassian and Standard Normal curves. I have used Desmos and Grapher (on mac) to draw these curves on this page.

But the question remains. Why is this such special curve? Exponential growth and exponential decay arise when rate of change in something depends on the current state of that something. For example drain rate of water from a container at any moment might depend on the quantity of the water in the container at that moment, giving rise to exponential decay. However the exponential rate of change can be modeled a using sum of infinite number of polynomial growth rates which do not depend on the underlying quantity.