Talk:Cumulative distribution function: Difference between revisions

Revision as of 18:08, 4 October 2021

Revision as of 19:20, 23 March 2023

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I went through and changed the notation <math>F_X(x)</math> to <math>F(x)</math> everywhere in the definition section to try to obtain notational consistency through the article, but the change was reverted by Nijdam with edit summary "Difference between cdf of X and just a cdf". But that conflicts with much notation in the article that uses F(x) for the cdf of X. In the Properties section:

:''<i>the CDF of ''X'' will be discontinuous at the points ''x''_''i'' and constant in between:''</i>

::<math>F(x) = \operatorname{P}(X\leq x) = \sum_{x_i \leq x} \operatorname{P}(X = x_i) = \sum_{x_i \leq x} p(x_i).</math>

:''<i>If the CDF ''F'' of ''X'' is [[continuous function|continuous]], then ''X'' is a [[continuous random variable]]; if furthermore ''F'' is [[absolute continuity|absolutely continuous]], then there exists a [[Lebesgue integral|Lebesgue-integrable]] function ''f''(''x'') such that''</i>

::<math>F(b)-F(a) = \operatorname{P}(a< X\leq b) = \int_a^b f(x)\,dx</math>

:''<i>for all real numbers ''a'' and ''b''. The function ''f'' is equal to the [[derivative]] of ''F'' [[almost everywhere]], and it is called the [[probability density function]] of the distribution of ''X''.''</i>

In the Examples section:

:''<i>As an example, suppose ''X'' is [[uniform distribution (continuous)|uniformly distributed]] on the unit interval [0,&nbsp;1]. Then the CDF of X is given by''</i>

::<math>F(x) = \begin{cases}

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== Continuous Random Variables ==

Elsewhere on Wikipedia, and in many published books, a continuous random variable has an ''absolutely'' continuous c.d.f., not merely continuous as stated in the properties section. I suggest that this page should also state that the c.d.f. is absolutely continuous so that there is a p.d.f. [[User:Paulruud|Paulruud]] ([[User talk:Paulruud|talk]]) 17:26, 30 March 2015 (UTC)<small><span class="autosigned"></small>

== Definition as expectation value ==

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I found this in the introduction of [[Characteristic function (probability theory)|Characteristic function]]:

''The characteristic function provides an alternative way for describing a [[random variable]]. Similarly to the [[cumulative distribution function]]''

:<math>F_X(x) = \operatorname{E} \left [\mathbf{1}_{\{X\leq x\}} \right],</math>

''<i>( where '''1'''_{{''X ≤ x''}} is the [[indicator function]] — it is equal to 1 when {{nowrap|''X ≤ x''}}, and zero otherwise), which completely determines behavior and properties of the probability distribution of the random variable ''X'', the '''characteristic function'''''</i>

: <math> \varphi_X(t) = \operatorname{E} \left [ e^{itX} \right ]</math>