Editing Probability distribution (section)

== Terminology ==

Some key concepts and terms, widely used in the literature on the topic of probability distributions, are listed below.<ref name=":02" />

=== Basic terms ===
*''[[Random variable]]'': takes values from a sample space; probabilities describe which values and set of values are more likely taken.
*''[[Event (probability theory)|Event]]'': set of possible values (outcomes) of a random variable that occurs with a certain probability.
*''[[Probability measure|Probability function]]'' or ''probability measure'': describes the probability <math>P(X \in E)</math> that the event <math>E,</math> occurs.<ref name='vapnik'>Chapters 1 and 2 of {{harvp|Vapnik|1998}}</ref>
*''[[Cumulative distribution function]]'': function evaluating the [[probability]] that <math>X</math> will take a value less than or equal to <math>x</math> for a random variable (only for real-valued random variables).
*''[[Quantile function]]'': the inverse of the cumulative distribution function. Gives <math>x</math> such that, with probability <math>q</math>, <math>X</math> will not exceed <math>x</math>.

=== Discrete probability distributions ===
*'''Discrete probability distribution''': for many random variables with finitely or countably infinitely many values.
*''[[Probability mass function]]'' (''pmf''): function that gives the probability that a discrete random variable is equal to some value.
*''[[Frequency distribution]]'': a table that displays the frequency of various outcomes {{em|in a sample}}.
*''[[Relative frequency]] distribution'': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] (i.e. sample size).
*''[[Categorical distribution]]'': for discrete random variables with a finite set of values.

=== Absolutely continuous probability distributions ===
*'''Absolutely continuous probability distribution''': for many random variables with uncountably many values.
*''[[Probability density function]]'' (''pdf'') or ''probability density'': function whose value at any given sample (or point) in the [[sample space]] (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would equal that sample.

=== Related terms ===
*[[Support (mathematics)|''Support'']]: set of values that can be assumed with non-zero probability (or probability density in the case of a continuous distribution) by the random variable. For a random variable <math>X</math>, it is sometimes denoted as <math>R_X</math>.
*'''Tail''':<ref name='tail'>More information and examples can be found in the articles [[Heavy-tailed distribution]], [[Long-tailed distribution]], [[fat-tailed distribution]]</ref> the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form <math>X > a</math>, <math>X < b</math> or a union thereof.
*'''Head''':<ref name='tail' /> the region where the pmf or pdf is relatively high. Usually has the form <math>a < X < b</math>.
*''[[Expected value]]'' or ''mean'': the [[weighted average]] of the possible values, using their probabilities as their weights; or the continuous analog thereof.
*''[[Median]]'': the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
*[[Mode (statistics)|''Mode'']]: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
*''[[Quantile]]'': the q-quantile is the value <math>x</math> such that <math>P(X < x) = q</math>.
*''[[Variance]]'': the second moment of the pmf or pdf about the mean; an important measure of the [[Statistical dispersion|dispersion]] of the distribution.
*''[[Standard deviation]]'': the square root of the variance, and hence another measure of dispersion.
*[[Symmetric probability distribution|''Symmetry'']]: a property of some distributions in which the portion of the distribution to the left of a specific value (usually the median) is a mirror image of the portion to its right.
*''[[Skewness]]'': a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third [[standardized moment]] of the distribution.
*''[[Kurtosis]]'': a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.