Symmetric probability distribution

In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function or probability mass function is reflected around a vertical line at some value of the random variable represented by the distribution. This vertical line is the line of symmetry of the distribution. Thus the probability of being any given distance on one side of the value about which symmetry occurs is the same as the probability of being the same distance on the other side of that value.

Formal definition

A probability distribution is said to be symmetric if and only if there exists a value ${\displaystyle x_{0}}$ such that

${\displaystyle f(x_{0}-\delta )=f(x_{0}+\delta )}$ for all real numbers ${\displaystyle \delta ,}$

where f is the probability density function if the distribution is continuous or the probability mass function if the distribution is discrete.

For continuous symmetric spherical, Mir M. Ali gave the following definition. Let ${\displaystyle {\mathcal {F}}}$denote the class of spherically symmetric distributions of the absolutely continuous type in the n-dimensional Euclidean space having joint density of the form ${\displaystyle f(x_{1},x_{2},\dots ,x_{n})=g(x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2})}$inside a sphere with center at the origin with a prescribed radius which may be finite or infinite and zero elsewhere.[1]

Properties

• The median and the mean (if it exists) of a symmetric distribution both occur at the point ${\displaystyle x_{0}}$ about which the symmetry occurs.
• If a symmetric distribution is unimodal, the mode coincides with the median and mean.
• All odd central moments of a symmetric distribution equal zero (if they exist), because in the calculation of such moments the negative terms arising from negative deviations from ${\displaystyle x_{0}}$ exactly balance the positive terms arising from equal positive deviations from ${\displaystyle x_{0}}$.
• Every measure of skewness equals zero for a symmetric distribution.

Probability density function

Typically a symmetric continuous distribution's probability density function contains the index value ${\displaystyle x}$ only in the context of a term ${\displaystyle (x-x_{0})^{2k}}$ where ${\displaystyle k}$ is some positive integer (usually 1). This quadratic or other even-powered term takes on the same value for ${\displaystyle x=x_{0}-\delta }$ as for ${\displaystyle x=x_{0}+\delta }$, giving symmetry about ${\displaystyle x_{0}}$. Sometimes the density function contains the term ${\displaystyle |x-x_{0}|}$, which also shows symmetry about ${\displaystyle x_{0}.}$

Unimodal case

Partial list of examples

The following distributions are symmetric for all parametrizations. (Many other distributions are symmetric for a particular parametrization.)

• Arcsine distribution
• Bates distribution
• Cauchy distribution
• Champernowne distribution
• Continuous uniform distribution
• Degenerate distribution
• Discrete uniform distribution
• Elliptical distributions
• Gaussian q-distribution
• Generalized normal distribution
• Hyperbolic secant distribution
• Irwin–Hall distribution
• Laplace distribution
• Logistic distribution
• Normal distribution
• Normal-exponential-gamma distribution
• Rademacher distribution
• Raised cosine distribution
• Student's t distribution
• Tukey lambda distribution
• U-quadratic distribution
• Voigt distribution
• von Mises distribution
• Wigner semicircle distribution

References

1. Ali, Mir M. (1980). "Characterization of the Normal Distribution Among the Continuous Symmetric Spherical Class". Journal of the Royal Statistical Society. Series B (Methodological). 42 (2): 162–164. doi:10.1111/j.2517-6161.1980.tb01113.x. JSTOR 2984955.