Definition Density function

A density function (also known as a probability density function) describes the probability that a random variable will appear as a certain characteristic value. However, this only applies to instances with discrete attributes. For steady attributes, probabilities are determined by the distribution function, meaning no determinations about the characteristic value can be made leveraging the density function.

Important discrete distribution types are binomial, hypergeometric, and Poisson. The famous bell-shaped curve of normal distribution, which is also known as the Gaussian curve, is a density function (and not, as is often said, a distribution function).

Here is a (simplified) example which nicely illustrates the benefits of the density function: In a survey of 10,000 people, all participants are asked how much money they have at the end of the month (after taxes, rent, and other expenses). The result is shown in a density function. By looking at the X-axis and selecting a specific value, such as $123, it is possible to calcuate the area between the X-axis and the density function to the left of this area.This area illustrates the proportion of people who have less than the amount of $123 left over at the end of the month. For this purpose, the size of the area is divided by the total size of the area between the density function and the X-axis. This can be repeated for any value of the X-axis.