Absolutely continuous probability distribution

The absolutely continuous (probability) distributions , also called absolutely continuous probability measures , are a special class of probability measures in stochastics . They are characterized by the fact that they can be defined or represented using an integral and a probability density function.

Although they are closely related to the continuous probability distributions , they are not identical to them.

definition

A probability measure on is absolutely continuous if absolutely continuous of respect Lebesgue measure is. This means that every - zero set is also a - zero set . ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle P}$

According to Radon-Nikodým's theorem , this is equivalent to having a probability density function . That means it applies to everyone with ${\displaystyle P}$ ${\displaystyle f_{P}}$ ${\displaystyle [a,b]}$ ${\displaystyle a<b}$

${\displaystyle P([a,b])=\int _{a}^{b}f_{P}(x)\mathrm {d} x}$.

comment

Strictly speaking, one would have to define the probability density function in such a way that it is clear that it is a density with respect to the Lebesgue measure. In stochastics, however, densities with regard to measures other than the Lebesgue measure are rare, which is why they are often omitted.

Strictly speaking, the integral is a Lebesgue integral . However, this is often replaced by a Riemann integral , as is the case here , and instead of is written . ${\displaystyle \mathrm {d} x}$ ${\displaystyle \mathrm {d} \lambda (x)}$

Delimitation from the continuous probability distributions

As a continuous probability distributions those probability distributions are called, which is a continuous distribution function possess. Applied to the extent means that the continuous probability distributions atomlos are, so no single points with possess. ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle P(\{x\})>0}$

After the Lebesgue decomposition , atomic measures can be further split:

• In an absolutely constant proportion. This corresponds to the absolutely continuous probability distributions.
• In a singular part. This corresponds to the continuously singular probability distributions .

Thus every absolutely continuous probability distribution is always a continuous probability distribution. But not every continuous probability distribution is an absolutely continuous probability distribution. An example of this is the Cantor distribution : its distribution function is continuous, but it has no probability density function.

literature

• 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 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ^ Schmidt: Measure and probability. 2011, p. 255.
2. ^ Georgii: Stochastics. 2009, p. 242.