Density function

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A density function , or density for short , is a special real-valued function that occurs mainly in the mathematical sub-areas of stochastics and measure theory . There, density functions are used to construct dimensions or signed dimensions using integrals .

The best-known example of density functions are the probability density functions from probability theory . With their help, many probability measures can be constructed without having to resort to more deeply-based methods and structures.


Let a dimension space and a positive, quasi-integrable function be given


Then you can get through

for all

define a measure. The function is then called the density function of the measure.

Are reversed and dimensions on and is

for a positive quasi-integrable function and all ,

this is the name of the density function of measure with respect to measure . The function is then also referred to as the Radon-Nikodým density or Radon-Nikodým derivative and noted as being.

The definition for signed dimensions is identical in both cases, only the positivity of the quasi-integrable functions is dropped.


Probability density functions

Typical examples of density functions are probability density functions . These are density functions with respect to the Lebesgue measure or the Lebesgue integral , in which the measure of the base space is one. Specifying such a function is a simple way of using probability measures

define. Probability measures that can be defined in this way are called absolutely continuous probability measures . They allow an elementary access to probability theory, often the Lebesgue integral is not used and the Riemann integral is used instead . Then the notation is used instead of .

Counting densities

Counting densities , also called probability functions, are another example of density functions . In the simplest case, you assign a positive number to every natural number:


The function values ​​add up to one and thus over define

a discrete probability distribution . If we choose as a measure the counting measure on , so


Counting densities are therefore density functions with regard to the counting measure.


By definition, every positive quasi-integrable function can be used in combination with a measure to define a further measure and thus be declared a density function.

If, however, two measures are given, the question arises whether a density function has or vice versa. Radon-Nikodým's theorem answers this question :

If σ-finite and is absolutely continuous with respect to , then has a density function with respect to .

See also


Individual evidence

  1. Klenke: Probability Theory. 2013, p. 159.