Symmetric probability distribution

In statistics, a symmetric distribution or 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 $x_{0}$ such that

f(x_{0}-\delta )=f(x_{0}+\delta )

for all real numbers

\delta ,

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

Properties

The median and the mean (if it exists) of a symmetric distribution both occur at the point $x_{0}$ about which the symmetry occurs.

All odd central moments of a symmetric distribution equal zero, because in the calculation of such moments the negative terms arising from negative deviations from the mean exactly balance the positive terms arising from equal positive deviations from the mean.

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

Partial list of examples

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