Cumulative distribution function: Difference between revisions

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If treating several random variables <math>X,Y,\ldots</math> etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital <math>F</math> for a cumulative distribution function, in contrast to the lower-case <math>f</math> used for [[probability density function]]s and [[probability mass function]]s. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the [[normal distribution]].
 
The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating<ref>{{Cite book|title=Applied Statistics and Probability for Engineers|last1=Montgomery|first1=Douglas C.|last2=Runger|first2=George C.|publisher=John Wiley & Sons, Inc.|year=2003|isbn=0-471-20454-4|page=104|url=http://www.um.edu.ar/math/montgomery.pdf}}</ref> using the [[Fundamental Theorem of Calculus]]; i.e. given <math>F(x)</math>,
 
: <math>f(x) = {dF(x) \over dx},</math> as long as the derivative exists.
[[Probability density function]] from the cumulative distribution function<ref>{{Cite book|title=Applied Statistics and Probability for Engineers|last=|first=|publisher=|year=|isbn=1119456266|location=|pages=70}}</ref>
 
as long as the derivative exists.
Given ''F''(''x''),
 
: <math>f(x) = {dF(x) \over dx},</math> as long as the derivative exists.
 
The CDF of a [[continuous random variable]] <math>X</math> can be expressed as the integral of its probability density function <math>f_X</math> as follows:<ref name="KunIlPark" />{{rp|p. 86}}