Gaussian Distribution

Introduction

The Gaussian distribution, also known as the normal distribution, is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is an exponent of a quadratic function. The parameters of the Gaussian distribution, namely the mean and variance, determine the shape and spread of the distribution.

A bell curve representing a Gaussian distribution

A bell curve representing a Gaussian distribution

Mathematical Description

The Gaussian distribution is mathematically described by the probability density function:

\[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{ - \frac{(x-\mu)^2}{2\sigma^2} } \]

where:

• \( \mu \) is the mean or expectation of the distribution (and also its median and mode),
• \( \sigma \) is the standard deviation, and
• \( \sigma^2 \) is the variance.

The factor \( \frac{1}{\sqrt{2\pi\sigma^2}} \) ensures that the total probability integrates to 1.

Properties

Symmetry

The Gaussian distribution is symmetric about its mean. This property is due to the quadratic form in the exponent of the density function. This symmetry leads to the well-known bell curve shape.

Mean, Median, and Mode

The mean, median, and mode of a Gaussian distribution are all equal and are located at the center of the distribution. This is a direct result of the symmetry of the distribution.

Variance and Standard Deviation

The variance and standard deviation of a Gaussian distribution measure the spread or dispersion of the distribution. A larger variance or standard deviation indicates a wider spread.

Skewness and Kurtosis

The skewness of a Gaussian distribution is zero, indicating no skew or asymmetry. The kurtosis of a Gaussian distribution is 3, indicating a relatively flat peak compared to other distributions.

Central Limit Theorem

The central limit theorem is a fundamental theorem in probability theory and statistics that states that the sum of a large number of independent and identically distributed random variables will approximately follow a Gaussian distribution, regardless of the shape of the original distribution. This theorem is the foundation for many statistical procedures and justifies the widespread use of the Gaussian distribution in science and engineering.

Applications

The Gaussian distribution is widely used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Its ubiquity is due to the central limit theorem. It is also used in signal processing, image processing, and data fitting.

See Also

• Central Limit Theorem
• Probability Density Function
• Standard Deviation
• Variance