Appendix

Special Issue- Gaussian Distribution

The Man Behind This

CHAPTER ONE

Why the Normal Distribution Is Everywhere

You know when you look at how tall people’re in a big group or you see the mistakes in a physics experiment or you check the daily changes in a stock index? That is when you see the distribution even if you do not know what it is called.

The normal distribution is everywhere in nature. The things people do. It is like a bell curve. A time ago a French mathematician named Abraham de Moivre saw this normal distribution in the 1730s. He probably thought he found something important, about the world. He really did find something important. The normal distribution is something you see all the time. That is why it is so interesting.

Phenomena that are normally distributed

Consider these amounts as mentioned below. They all follow a distribution or come close to it. These quantities are like height, weight and IQ scores. They all have a bell-curve shape. This means most values are, around the average and fewer are very high or very low. The normal distribution is also called a bell-curve. It helps us understand how these quantities are spread out.

The main idea here is that when you have something that is made up of small things that happen by chance and do not depend on each other the Central Limit Theorem comes into play. The Central Limit Theorem says that when you add up all these random things the result is usually close to a normal distribution. We will learn more, about the Central Limit Theorem in Chapter 9. The Central Limit Theorem is really important because it helps us understand how the normal distribution works.

KEY OBSERVATION

Not everything follows a Normal distribution. Income is heavily right-skewed I mean 
it is not spread out evenly. The number of wars per decade seems to follow is a 
Poisson distribution. It is crucial to know when normal distribution is applied 
and when it does not. This is just as important, as understanding the distribution 
itself.

The formula we are about to derive

The probability density function (PDF) of the normal distribution looks terrifying at first glance:

This formula contains e (Euler’s number), П (pie), and a complicated exponent. But each ingredient has a precise, beautiful reason to be there. By the end of this article, you will understand exactly why the formula has this form — and it will seem not just reasonable, but inevitable.

Our derivation follows three major steps:

CHAPTER TWO

Setting the Stage: The Dart Board Argument

When we start to figure this out we have a simple idea to begin with. Think about throwing darts at a target. You want to hit the center of the target.. Your throws are not perfect. Each dart lands somewhere random. However you throw the darts in a consistent way. So we can make some guesses about how far off your throws are, from the center. We are talking about the darts and the target and the darts are what we are focusing on. The target is important too because it is what we are aiming the darts at.

The key assumptions

We are going to make two assumptions about the dart-throwing model. The dart-throwing model has some assumptions that seem normal and they make sense in the real world but the dart-throwing model assumptions have some really big effects when we look at the math, behind the dart-throwing model.

WHY THESE ASSUMPTIONS?

These are not arbitrary. Independence of horizontal and vertical errors is 
natural for many physical processes — wind pushes in one direction while hand 
tremor affects the other. Isotropy says there is no preferred direction of 
error. Together, these two assumptions completely determine the functional 
form of the distribution.

We now derive the form of f from our two assumptions. This is the most mathematically demanding part of the derivation, but the steps follow naturally once you see the trick.

Setting up the functional equation

We have established that our two assumptions require:

Using the chain rule on the left side:

Using the chain rule on the left side:

In polar coordinates, x = rcosΘ and y = rsinΘ, so:

Substituting these into the differentiated equation:

Separating variables