What is Gaussian Distribution?
The Gaussian distribution, also known as the normal distribution, is a continuous probability distribution that is widely used in statistical modeling and Machine Learning. It is a bell-shaped curve that is symmetrical around its mean and is characterized by its mean and standard deviation.
A wide variety of real-world occurrences may be adequately modeled by the Gaussian distribution. Simply put, the central limit theorem is the mathematical embodiment of the intuitive reality that, when aggregated, numerous measurable quantities tend to be of the same values with just a few outliers.
It’s a graphical depiction of the Central Limit Theory (CLT) in action, showing how a huge number of data points cluster together . According to the CLT, as the number of values in a sum of random values rises, the distribution of those values will move toward a Gaussian. Numerous instances of real-world data like the ground state of a quantum harmonic oscillator and the distribution of demographic traits across populations, may be described by a Gaussian distribution.
Importance of Gaussian Distribution
This distribution’s importance lies in the simple fact that it describes the distribution of many naturally occurring phenomena such as the distribution of heights or IQ scores of people.
One of the main reasons is the central limit theorem which states that the sum of a large number of independent and identically distributed (IID) random variables will tend to be normally distributed, regardless of the distribution of the individual variables. This means that if you have a large sample of data, the distribution of the sample mean will be approximately normally distributed, even if the original data is not normally distributed.
- The Gaussian distribution is important because it is a common and useful model for data, and because it allows for efficient calculations in statistical analysis.
Another reason is its tractability. Many statistical methods and models are based on the assumption that the data is normally distributed, and this assumption allows for simpler and more efficient calculations. For example, if a dataset is normally distributed, you can use the standard normal distribution (a special case of the Gaussian distribution with a mean of 0 and standard deviation of 1) to calculate probabilities and make statistical inferences.
It is widely used in many fields including physics, engineering, economics, and biology to represent natural phenomena and to analyze data.
Gaussian Distribution formula
- f(x) = (1 / sqrt(2 * pi * sigma^2)) * exp(-((x – mu)^2) / (2 * sigma^2))
In this formula:
- X is a real number representing a possible value of a continuous random variable;
- mu is the mean of the distribution, and sigma is the standard deviation;
- (1 / sqrt(2 * pi * sigma^2))– is the normalization factor that ensures that the area under the curve of the distribution is equal to 1; and
- exp(-((x – mu)^2) / (2 * sigma^2))– is a bell-shaped curve that is centered at the mean, mu, and has a standard deviation of sigma.
The inverse Gaussian distribution
This is a continuous probability distribution that is often used to model the time required to complete a task or the distance traveled by a particle.
It is defined by this function:
- f(x) = (sqrt(lambda / (2 * pi * x^3))) * exp(-lambda * (x – mu)^2 / (2 * mu^2 * x))
In this formula:
- X is a positive real number representing a possible value of a continuous random variable;
- mu is the mean of the distribution, and lambda is a scale parameter.
- (sqrt(lambda / (2 * pi * x^3)))– is a normalization factor that ensures that the area under the curve of the distribution is equal to 1; and
- exp(-lambda * (x – mu)^2 / (2 * mu^2 * x))– is a bell-shaped curve that is centered at the mean, mu, and has a scale parameter of lambda.
The inverse Gaussian distribution has several properties that make it useful for modeling certain types of data. It has a heavier tail than the Gaussian distribution, which means that it is more likely to produce extreme values. It is also skewed to the right since the distribution is shifted to the right of the mean. This makes it useful for modeling data that has a long tail or is skewed to the right.
The Financial Market and the Gaussian Distribution
Asset prices and price behavior are both assumed to follow a normal distribution. Points in price history may be plotted by traders in an attempt to model the most recent price movement as a normal distribution. An asset is more likely to be over or undervalued the more its price deviates from the mean. Potential trades might be suggested by looking at the standard deviations. Because it is difficult to judge when to enter and quit a trade on longer time frames, most traders who engage in this practice limit their transactions to shorter intervals of time.
To a similar extent, several statistical theories seek to describe asset values based on the premise that they follow a normal distribution. In practice, price distributions have fat tails, hence their kurtosis is often bigger than 3. These assets’ prices have deviated more than three standard deviations from the mean, which is significantly out of line with what one would anticipate from a normally distributed data set. Past performance may not reliably foretell future results, even if an asset has followed a normal distribution over a lengthy period of time.