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. ${\displaystyle -x}$ ${\displaystyle x}$

definition

A probability distribution on is called symmetric (around zero) if the following applies to all : ${\displaystyle P}$ ${\displaystyle (\mathbb {R} ,{\mathcal {P}})}$ ${\displaystyle x\in \mathbb {R} }$

${\displaystyle P((-\infty ,-x])=P([x,+\infty ))}$

Analogously, a real-valued random variable is called symmetric (around zero) if the distribution of matches the distribution of , so it is true ${\displaystyle X}$ ${\displaystyle -X}$

${\displaystyle P_{X}=P_{-X}}$or . ${\displaystyle X{\stackrel {\mathcal {D}}{=}}-X}$

More generally, a probability measure is called symmetric if ${\displaystyle P}$ ${\displaystyle a}$

${\displaystyle P((-\infty ,a-x])=P([a+x,+\infty ))}$

applies to all , just as a real-valued random variable is called symmetric um if ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle a}$

${\displaystyle X-a{\stackrel {\mathcal {D}}{=}}-(X-a)}$

applies.

First examples

• The uniform distribution is symmetrical about its expected value.
• The normal distribution is symmetrical about its expected value . ${\displaystyle N(\mu ,\sigma )}$ ${\displaystyle \mu }$
• 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 ${\displaystyle F_{-}(x)}$ ${\displaystyle x}$

${\displaystyle F(-x)+F_{-}(x)=1}$

applies to all and symmetrically around if and only if ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle a}$

${\displaystyle F(a-x)+F_{-}(a+x)=1}$.

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. ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle a}$ ${\displaystyle x=a}$
• 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 . ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle a}$ ${\displaystyle x=a}$

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 ${\displaystyle X}$ ${\displaystyle a}$ ${\displaystyle (2k+1)}$

${\displaystyle \operatorname {E} ((X-a)^{2k+1})=0}$.

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

${\displaystyle \varphi _{X}(t)=\operatorname {E} (\cos(tX))}$.

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	${\displaystyle p={\tfrac {1}{2}}}$	${\displaystyle a={\tfrac {1}{2}}}$	For see Dirac distribution to 0 or 1 ${\displaystyle p\in \{0,1\}}$
Binomial distribution ${\displaystyle X\sim Bin(n,p)}$	${\displaystyle p={\tfrac {1}{2}}}$	${\displaystyle a={\frac {n}{2}}}$	Goes to or into the Dirac distribution , see symmetries there. ${\displaystyle p\in \{0,1\}}$ ${\displaystyle 0}$ ${\displaystyle n}$
Discrete equal distribution on ${\displaystyle \{r,r+1,\dots ,r+n\}}$	${\displaystyle r\in \mathbb {R} ,\,k\in \mathbb {N} }$	${\displaystyle a={\frac {2r+n}{2}}}$
Rademacher distribution	-	${\displaystyle a=0}$
Two-point distribution on ${\displaystyle \{c,d\}}$	${\displaystyle c,d\in \mathbb {R} ,\,p={\tfrac {1}{2}}}$	${\displaystyle a={\frac {c+d}{2}}}$	For a degenerate case, see Dirac distribution. ${\displaystyle p\in \{0,1\}}$
Absolutely continuous distributions
Normal distribution	${\displaystyle \mu \in \mathbb {R} ,\,\sigma ^{2}>0}$	${\displaystyle a=\mu }$
Constant equal distribution on ${\displaystyle [c,d]}$	-	${\displaystyle a={\frac {c+d}{2}}}$
Cauchy distribution	${\displaystyle t\in \mathbb {R} ,\,s>0}$	${\displaystyle a=t}$	Typical example of a symmetrical distribution with no expected value
Student's t-distribution	${\displaystyle n\in \mathbb {N} }$	${\displaystyle a=0}$
Beta distribution on ${\displaystyle (0,1)}$	${\displaystyle p=q}$	${\displaystyle a={\tfrac {1}{2}}}$
Arcsin distribution	-	${\displaystyle a={\tfrac {1}{2}}}$
Logistic distribution	${\displaystyle \alpha \in \mathbb {R} ,\,\beta >0}$	${\displaystyle a=\alpha }$
Continuously singular distributions and degenerate distributions
Cantor distribution	-	${\displaystyle a={\tfrac {1}{2}}}$
Dirac distribution ${\displaystyle \delta _{x}}$	${\displaystyle x\in \mathbb {R} }$	${\displaystyle a=x}$

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 .