Symmetrical probability distribution
In stochastics, special probability distributions on real numbers are called symmetrical (probability) distributions . They are characterized by the fact that (in the simplest case) the probability of receiving a value smaller than is always the same as the probability of receiving a value greater than . If a random variable has a symmetrical distribution, it is also called a symmetrical random variable.
definition
A probability distribution on is called symmetric (around zero) if the following applies to all :
Analogously, a real-valued random variable is called symmetric (around zero) if the distribution of matches the distribution of , so it is true
- or .
More generally, a probability measure is called symmetric if
applies to all , just as a real-valued random variable is called symmetric um if
applies.
First examples
- The uniform distribution is symmetrical about its expected value.
- The normal distribution is symmetrical about its expected value .
- The exponential distribution or the Poisson distribution are not symmetrical, i.e. not symmetrical about any point .
properties
Characterization by the distribution function
The symmetry of a random variable / distribution can also be characterized or defined via its distribution function . If the left-hand limit value is used at the point , then the distribution or random variable is symmetrical about zero if and only if
applies to all and symmetrically around if and only if
- .
Probability density functions and probability functions
The symmetry of a probability distribution can also be defined directly via the probability (density) functions of the distribution:
- Is an absolutely continuous distribution , then if and symmetrical about , when the probability density function axisymmetrical with respect. The axis is.
- If there is a discrete distribution on the real numbers, then it is symmetrical about if and only if the probability function is axially symmetrical with respect to the axis .
Median and moments
The center of symmetry always agrees with a median , as does the expected value if this exists. However, this does not always have to be the case with symmetrical probability distributions, as the standard Cauchy distribution shows: It is symmetrical around zero, but its expected value does not exist.
In general: if it is a random variable that is symmetrical about symmetry and if its -th moment exists , then it is
- .
Characteristic functions
The characteristic function of a probability distribution is real-valued if and only if the distribution is symmetric about zero, and then applies
- .
Furthermore, Pólya's theorem enables the construction of functions that are always characteristic functions of a distribution symmetrical about zero.
More symmetrical distributions
distribution | for parameter selection | Symmetrical around | comment |
---|---|---|---|
Discrete distributions | |||
Bernoulli distribution | For see Dirac distribution to 0 or 1 | ||
Binomial distribution | Goes to or into the Dirac distribution , see symmetries there. | ||
Discrete equal distribution on | |||
Rademacher distribution | - | ||
Two-point distribution on | For a degenerate case, see Dirac distribution. | ||
Absolutely continuous distributions | |||
Normal distribution | |||
Constant equal distribution on | - | ||
Cauchy distribution | Typical example of a symmetrical distribution with no expected value | ||
Student's t-distribution | |||
Beta distribution on | |||
Arcsin distribution | - | ||
Logistic distribution | |||
Continuously singular distributions and degenerate distributions | |||
Cantor distribution | - | ||
Dirac distribution |
literature
- Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin / Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 38 , doi : 10.1007 / 978-3-642-17261-8 .
- Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin / Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 244-245 , doi : 10.1007 / 978-3-642-45387-8 .
- 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 .