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# Gaussian Distribution

## Understanding Gaussian Distribution

Gaussian Distribution, also known as Normal Distribution, holds a significant place in statistical modelling and machine learning. This distribution, characterized by its mean and standard deviation, takes the shape of a bell curve symmetrical around its mean.

The Gaussian Distribution is compelling due to the Central Limit Theory (CLT), which suggests that an aggregate of many quantifiable measurements usually cluster around the same values, barring a few anomalies. With an increase in aggregated values, the distribution of these values tends to shift towards Gaussian. It is common to find instances of real-world data, such as quantum harmonic oscillator's ground state or demographic characteristics distribution, modelled adequately by the Gaussian Distribution.

## Importance of Gaussian Distribution

The significant contribution of Gaussian Distribution lies in its ability to detail the distribution of many naturally occurring events like the height distribution or IQ scores of individuals. According to the Central Limit Theory, even if the primary data isn't normally distributed, the sum of an extensive collection of independently and uniformly distributed (IID) random variables will follow a Gaussian Distribution.

Gaussian Distribution is a popular choice for data modellers because it makes for an efficient calculation model in statistical analysis. More so, several statistical methods are grounded on the assumption of normally distributed data, simplifying calculations. If the data is normally distributed, the standard normal distribution can be used to compute probabilities and draw statistical inferences.

Gaussian Distribution is widely applied across various domains such as physics, engineering, biology, and economics for representing natural phenomena and data analysis.

## Gaussian Distribution Formula

The Gaussian Distribution formula is given as:

f(x) = (1 / sqrt(2 * pi * sigma^2)) * exp(-((x – mu)^2) / (2 * sigma^2)).

In this formula, 'x' is a real number symbolizing a constant random variable's potential value; 'mu' is the distribution mean and 'sigma' the standard deviation.

Also, there's the Inverse Gaussian Distribution that is a continuous probability distribution regularly used to model the time taken to finish a task or distance covered by a particle. This model is employed when the data set displays attributes like heavier tail or right skewing.

## Gaussian Distribution in Financial Markets

In financial markets, it is assumed that asset prices follow a Gaussian Distribution. Traders employ models based on the Gaussian Distribution to predict potential trades, analyze price deviations, and evaluate asset over/under valuation. However, the crucial take-away is that even if an asset has observed a normal distribution for an extended period, it doesn't secure its future performance reliability.

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