\[f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]
Variables
fprobability density
xrandom variable value
mumean of the distribution
sigmastandard deviation
Description
What is this formula?
This formula calculates the probability density of a normal distribution at a given value x. The normal distribution is one of the most important probability distributions in statistics, engineering, physics, finance, and data science.
When to use it
Use this formula when a variable follows a Gaussian or normal distribution and you need to evaluate the relative likelihood of observing a specific value.
Example
If the mean is 100 and the standard deviation is 15, the formula can estimate the density around values such as 90, 100, or 120.
Applications
Quality control, measurement uncertainty, signal processing, machine learning, financial modeling, natural phenomena analysis, and statistical inference.