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Symmetric probability distribution

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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.

Contents

Formal definitionEdit

A probability distribution is said to be symmetric if and only if there exists a value   such that

  for all real numbers  

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

PropertiesEdit

  • The median and the mean (if it exists) of a symmetric distribution both occur at the point   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   exactly balance the positive terms arising from equal positive deviations from  .
  • Every measure of skewness equals zero for a symmetric distribution.

Probability density functionEdit

Typically a symmetric continuous distribution's probability density function contains the index value   only in the context of a term   where   is some positive integer (usually 1). This quadratic or other even-powered term takes on the same value for   as for  , giving symmetry about  . Sometimes the density function contains the term  , which also shows symmetry about  

Unimodal caseEdit

Partial list of examplesEdit