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.
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 , 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
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 .
- Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
- Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
- David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
- Klenke: Probability Theory. 2013, p. 159.