Understanding the GAUSSIAN Distribution

Understanding the GAUSSIAN Distribution

Randomness is, present in our reality, we are used to take it for granted. The most of the phenomena which surround us has been generated by random processes. Hence, our brain are very good at recognise these random patterns. and is even better at spotting phenomena that should being random but they was actually aren ' t. And this was when problems arise. Most software such as Unity or gamemaker simply lack the tools to generate realistic random numbers. This tutorial'll introduce the Gaussian distribution, which plays a fundamental role in statistics since it's at the He Art of many random phenomena in we everyday life.

Introduction

Let's imagine you want to generate some the random points on a plane. They can enemies, trees, or whichever other entity you might thing of. The easiest-to-do it in Unity is:

1234	Vector3 position = new Vector3(); position. X = Random. Range(min,max), position. Y = Random. Range(min,max); transform. Position = position;

Using Random.  Range would produce points distributed like in the blue box below. Some points might is closer than others, but globally they is all spread all over the place with the same density. We find approximately as many points on the left as there is on the right.

Many natural behaviours don ' t follow this distribution. They is, instead, similar to the diagram on the Left:these phenomena is Gaussian distributed. Thumb Rule:when you had a natural phenomenon which should be around a certain value, the Gaussian distribution Could be the the-go. For instance:

• Damage: The amount of Damage an enemy or a weapon inflicts;
• particle density: The amount of particles (sparkles, dust, ...) around a particular object;
• Grass and trees: How Grass and trees is distributed in a biome; For instance, the position of plants near a lake, or the scatter or rocks around a mountain;
• Enemy Generation: If you want to generate enemies with random stats, can design a "average" Enemy and use th e Gaussian distribution to get natural variations out of it.

This tutorial would explain what a Gaussian distribution exactly are, and why it appears in all the above mentioned Phenomen A.

Understanding Uniform Distributions

When you ' re throwing a dice, there are one chance out of six to get a 6. Incidentally, every face of the dice also have the same chance. statistically speaking, throwing a dice samples f Rom a  uniform discrete distribution   (left).  every uniform distribution can be intuitively represented with a dice with  faces. Each face  has the same probability of being chosen . A function Such as random. Range, instead, returns values which are  continuously uniformly distributed   (right) Over A particular range (typically, between 0 and 1).

In many cases, uniform distributions is a good choice. Choosing a random card from a deck, for instance, can is modelled perfectly with random. Range.

What's a Gaussian distribution

There is phenomena in the natural domain which don ' t follow a uniform distribution. If you measure the height of the "people in a", you'll find that certain ranges occur more often than others. The majority of people'll has a similar height, while extreme tall or short people is rare to find. If you randomly choose a, the he height is likely to being close to the average height. These phenomena typically follow a distribution called the Gaussian (or normal) distribution. In a Gaussian distribution the probability of a given value to occur are given by:

 

If A uniform distribution is fully defined with its parameter, a Gaussian distribution are defined by both parameters and , namely the mean and the variance. The mean translates the curve left or right and centring it on the value which are expected to occur most frequently. The standard deviation, as the "name suggests", indicates how easy was to deviate from the mean.

When a variable was generated by a phenomenon which was Gaussian distributed, it is usually indicated as:

 

Converging to a Gaussian distribution

Surprisingly enough, the equation for a Gaussian distribution can is derived from a uniform distribution. Despite looking quite different, they is deeply connected. Now let's imagine a scenario in which a drunk man have to walk straight down a line. At every step, he had a 50% chance of moving left, and another 50% chance of moving right. Where is the most likely to find the drunk mans after 5 step? And after 100?

Since every step has the same probability and all of the above paths is equally likely to occur. Always going left is as likely as alternating left and right for the entire time. However, there is only one path which leads to he extreme left and while there was many more paths leading to the centre (MO Re details here). For this reason, the drunk mans is expected to stay closer to the centre. Have enough drunk men and enough time to walk, their final positions always approximate a Gaussian curve.

This concept can is explored without using actual drunk men. In the 19th century, Francis Galton came up with a device called beans Machine:an old fashionedpachinko which all oWS for balls to naturally arrange themselves into the typical Gaussian bell.

This was related with the idea behind the central limit theorem; After a sufficiently large number of independent, well defined trials, results should approximate a Gaussian curve, regard Less the underlying distribution of the original experiment.

Deriving the Gaussian distribution

If We look in The bean machine, we can ask a very simple question:what are the probability for a ball to en D up in a certain column? The answer depends on the number of right (or left) turns the ball makes. It is important to notice that the order doesn ' t really matter:both (left, left, right) and (right, left, left) lead&nbsp ; To the same column. And since there is a 50% change of going left or right at every turn, the question Becomes:how many left turns   is the ball making over  iterations (in the example above:  left turns over , iterations)? This can is calculated considering the chance Of turning Left  times, with the chance of turning righ t  times: . This form, however, accounts for only a single path:the one with  left turns followed by & Nbsp;right turns. We need to take into account all the possible permutations since they all leads to the same resUlt. without going too much into details, the number of permutations are described by the expression :

 

This is known as the binominal distribution and it answers the question of what likely is to obtain successes out of Inde Pendent experiments, each one with the same probability.

Even so, it still doesn ' t look very Gaussian @ all. The Bring to the infinity, switching from a discrete to a continuous distribution. In order to does, we first need to expand the binomial coefficient using its factorial form:

 

Then, factorial terms should is approximated using the Stirling ' s formula:

 

The rest of the derivation is mostly mechanic and incredibly tedious;  If you is interested, you can find it here. As a result we obtain:

 

With and.

Conclusion

This loosely explains why the majority of recurring, independent "natural" phenomena is, indeed, normally distributed. We are so surrounded by this distribution, We brain is incredibly good at recognise patterns which don ' t follow it. This is the reason so, especially in games, was important to understand that some aspects must follow a normal distributio N in order to is believable.

In the next post I'll explore how to generate Gaussian distributed numbers, and how they can is used safely in your game.

• Part 1: Understanding the Gaussian distribution
• Part 2: What to generate Gaussian distributed numbers

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Understanding the GAUSSIAN Distribution