Gaussian Distribution

The Gaussian distribution (also called the normal distribution) is the most important probability distribution in statistics and physics.

Probability Density Function

$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) $$

where:

• $\mu$ = mean (center of the distribution)
• $\sigma$ = standard deviation (spread)
• $\sigma^2$ = variance

The standard normal distribution has $\mu = 0$ and $\sigma = 1$.

Properties

Property	Value
Mean	$\mu$
Variance	$\sigma^2$
Skewness	0
Kurtosis	3

68-95-99.7 Rule: Approximately 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean.

Why It’s Everywhere

Central Limit Theorem

The sum of many independent random variables tends toward a Gaussian distribution, regardless of the original distributions. This explains why Gaussian distributions appear in:

• Measurement errors
• Thermal noise
• Brownian motion

Maximum Entropy

Among all distributions with a given mean and variance, the Gaussian has maximum entropy — it assumes the least additional structure.

Fourier Transform

The Gaussian is an eigenfunction of the Fourier transform:

$$ \mathcal{F}\{e^{-\pi x^2}\} = e^{-\pi f^2} $$

A Gaussian transforms into another Gaussian.

Multidimensional Gaussian

In $n$ dimensions with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$:

$$ f(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}|\boldsymbol{\Sigma}|^{1/2}} \exp\left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\mathbf{x}-\boldsymbol{\mu})\right) $$

Applications

• Statistics and hypothesis testing
• Machine learning (Gaussian processes, GMMs)
• Physics (thermal distributions, quantum mechanics)
• Signal processing (noise models)
• Finance (Black-Scholes model)

See Also

• Fourier transform