Editing Probability density function (section)

{{Short description|Description of continuous random distribution}}
{{Use American English|date = January 2019}}
{{citations needed|date=June 2022}}

[[Image:Boxplot vs PDF.svg|thumb|350px|[[Box plot]] and probability density function of a [[normal distribution]] {{math|''N''(0,&thinsp;''σ''<sup>2</sup>)}}.]]
[[Image:visualisation_mode_median_mean.svg|thumb|150px|Geometric visualisation of the [[mode (statistics)|mode]], [[median (statistics)|median]] and [[mean (statistics)|mean]] of an arbitrary unimodal probability density function.<ref>{{cite web|title=AP Statistics Review - Density Curves and the Normal Distributions|url=http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions | access-date=16 March 2015| archive-url=https://web.archive.org/web/20150402183703/http://apstatsreview.tumblr.com/post/50058615236/density-curves-and-the-normal-distributions | archive-date=2 April 2015| url-status=dead}}</ref>]]

In [[probability theory]], a '''probability density function''' ('''PDF'''), '''density function''', or '''density''' of an [[absolutely continuous random variable]], is a [[Function (mathematics)|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 be equal to that sample.<ref>{{cite book| chapter-url=https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf |archive-url=https://web.archive.org/web/20030425090244/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter4.pdf |archive-date=2003-04-25 |url-status=live| chapter=Conditional Probability - Discrete Conditional| last1=Grinstead|first1=Charles M.| last2=Snell|first2=J. Laurie| publisher=Orange Grove Texts| isbn=978-1616100469 | title=Grinstead & Snell's Introduction to Probability| date=2009| access-date=2019-07-25}}</ref><ref>{{Cite web|title=probability - Is a uniformly random number over the real line a valid distribution?| url=https://stats.stackexchange.com/q/541479 |access-date=2021-10-06| website=Cross Validated}}</ref> Probability density  is the  probability per unit length,  in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.

More precisely, the PDF is used to specify the probability of the [[random variable]] falling ''within a particular range of values'', as opposed to taking on any one value. This probability is given by the [[integral]] of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and the area under the entire curve is equal to 1.

The terms ''probability distribution function'' and ''probability function'' have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the [[probability distribution]] is defined as a function over general sets of values or it may refer to the [[cumulative distribution function]], or it may be a [[probability mass function]] (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion.<ref>Ord, J.K. (1972) ''Families of Frequency Distributions'', Griffin. {{isbn|0-85264-137-0}} (for example, Table 5.1 and Example 5.4)</ref> In general though, the PMF is used in the context of [[Continuous or discrete variable#Discrete variable|discrete random variables]] (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.