Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is a continuous probability distribution commonly used in statistical analysis. It is defined by the mean (μ) and the standard deviation (σ).

The PDF of the Gaussian distribution is given by:

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

where $x$ is an actual number, $\mu$ is the mean of the distribution, $\sigma$ is the standard deviation, $\pi$ is the mathematical constant π, and $e$ is the mathematical constant e.

The Gaussian distribution is symmetric about the mean, with the highest probability density occurring at the mean. As the standard deviation increases, the distribution becomes more expansive and flatter.

One of the critical properties of the Gaussian distribution is that about 68% of the observations fall within one standard deviation of the mean, about 95% of the observations fall within two standard deviations of the mean, and about 99.7% of the observations fall within three standard deviations of the mean.

The Gaussian distribution is widely used in various fields, such as physics, engineering, economics, and social sciences, due to its flexibility and ability to model many real-world phenomena. It is also a foundational concept in statistical inference, which involves making inferences about a population based on a sample of data.