Continuous probability distribution

The steady (probability) distributions , also called diffuse or atomless (probability) distributions or probability measures, are a large class of frequently occurring probability distributions on real numbers in stochastics . They are characterized by the fact that no isolated point is assigned a high probability. In this respect, they form the counterpart to the discrete probability distributions .

The continuous distributions are closely related to the absolutely continuous distributions , but not identical to them. They should therefore not be confused.

definition

A probability distribution on the real numbers is given , provided with Borel's σ-algebra . ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$

Then a continuous probability distribution is called if the distribution function of is continuous . ${\displaystyle P}$ ${\displaystyle F_{P}}$ ${\displaystyle P}$

Equivalent to this is that is atomless . That means there doesn't exist , so that is. ${\displaystyle P}$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle P(\{x\})>0}$

Further subdivision

According to the representation theorem, every continuous probability distribution can be further broken down into

• An absolutely continuous probability distribution for which a probability density function exists.
• A continuously singular probability distribution , the derivative of which vanishes almost everywhere .

Delimitation from the absolutely continuous probability distributions

As mentioned above, every absolutely continuous probability distribution is also always a continuous probability distribution. The converse does not apply in general, as the pathological example of the continuously singular Cantor distribution shows.

Thus, absolutely continuous and continuous probability distributions should not be confused. Due to the probability density function alone, the handling of absolutely continuous probability distributions is much easier than that of continuous ones.

literature

• 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 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .

Individual evidence

1. ^ Georgii: Stochastics. 2009, p. 242.