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Gaussian Distribution

What is a Gaussian Distribution?

The Gaussian distribution, normal distribution, or bell curve, is a probability distribution which accurately models a large number of phenomena in the world. Intuitively, it is the mathematical representation of the general truth that many measurable quantities, when taking in aggregate tend to be of the similar values with only a few outliers which is to say that many phenomena follow the central limit theorem.

Basically, it is the mathematical representation of how a large number of items follow the Central Limit Theory (CLT). The CLT says that , under mild conditions, the (normalized) sum of random values will tend to a gaussian distribution as the number of values in the sum increases. A Gaussian distribution can describe many examples of real-world data such as the ground state of a quantum harmonic oscillator or the distribution of demographic characteristics in populations.

A Tool for Inference

One useful fact about the ‘center heavy’ Gaussian is that it easily permits the definition of the standard deviation which is a quantity that describes where the majority of a sample set lies. 68% of data in a Gaussian falls within 1 standard deviation from the mean. 95% of the data may be found within 2 standard deviations and 99.7% of all data within 3 standard deviations. 

Occurrences of Gaussian Distribution in Nature and Machine Learning

The Gaussian distribution occurs in many physical phenomena such as the probability density function of a ground state in a quantum harmonic oscillator. Any particle undergoing diffusion (such as in a mixed liquid) may have its location modeled accurately as a Gaussian distribution as a function of time. Even sepal width of irises have been found to follow a Gaussian distribution.

A Gaussian process is any process in time or space that creates Gaussian distributions within its domain (time , space, etc). They may used to find non-linear regressions (one problem in machine learning) as well as to reduce dimensionality by identifying which dimensions of a dataset have larger variance and thus may contain more useful information. 

More Practical Uses of a Gaussian Distribution

Designing Standardized Tests – Standardized tests are designed so that test-taker scores fall within a Gaussian distribution.

Statistical Tests – Many statistical tests can be derived from a Gaussian distribution.

Quantum Mechanics – A Gaussian distribution can be used to describe the ground state of a quantum harmonic oscillator.