Talk:Cumulative distribution function: Difference between revisions

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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>

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I would like to know the full connection to measure theory. It seems if I have a probabilistic measure space <math>(A,\mathcal{A},P_A)</math> and a measurable function <math>X\colon A\to \mathbb{R}</math> it might be the pushforward measure <math>X_* P_A= P_A \circ X^{-1}</math> which I guess would be a mapping <math> (\mathbb{R}, \mathcal{B}(\mathbb{R}), ||\cdot||_2)\mapsto \mathbb{R}</math> which for some reason is then restricted to half open sets <math>F_X \overset{?}{:=} (X_* P_A)\restriction [-\infty , r)\times \mathbb{R}</math>. But this is original research and feels a bit patchy (unclear to me how to generalize to random variables with values in <math>\mathbb{R}^n</math>) so please if someone who has connected the dots could add the connection. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Rostspik|Rostspik]] ([[User talk:Rostspik#top|talk]] • [[Special:Contributions/Rostspik|contribs]]) 05:58, 27 April 2019 (UTC)</small> <!--Autosigned by SineBot-->

h <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2.183.8.104|2.183.8.104]] ([[User talk:2.183.8.104#top|talk]]) 05:39, 16 November 2019 (UTC)</small> <!--Autosigned by SineBot-->

== Check Required ==

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:{{re|Stobene45}} this is mostly a matter of Wikipedia's accepted style conventions. We don't artificially underline subheadings, but rather use the pre-defined styles generated by the staggered classes of headings; and while italics can be used for emphasis, this should be done very sparingly (see [[Wikipedia:Manual_of_Style/Text_formatting#Emphasis]]). As for the citations, references to Google results are not suitable anywhere on Wikipedia - the cited source must be the specific work used, not just a search result containing the work's title. Cheers! --<span style="font-family:Courier">[[User:Elmidae|Elmidae]]</span> <small>([[User talk:Elmidae|talk]] · [[Special:contributions/Elmidae|contribs]])</small> 16:26, 3 July 2021 (UTC)

== Mild error in Examples section ==

== uvhw.org ==

In the paragraph beginning "Suppose is [[Normal distribution|normal distributed]]. Then the CDF of  is given by" there is an error. The dummy variable "x" is used as the function argument F(x) rather than the upper limit of integration, "t". I will edit to make this F(t). [[Special:Contributions/2601:1C2:B00:C640:D541:DF77:FC10:64EF|2601:1C2:B00:C640:D541:DF77:FC10:64EF]] ([[User talk:2601:1C2:B00:C640:D541:DF77:FC10:64EF|talk]]) 23:56, 23 November 2024 (UTC)

:@[[User:Joe Gatt|Joe Gatt]] This edit and thread was created by me, I just hadn't logged in. Your edit was incorrect. I think you were confused by the notation - it's OK to have F(x) = int ^ x. What wouldn't be OK would be to have F(X) = int ^ X as a CDF definition (see [[Probability integral transform]]). Technically, X is a function of a probability space. Under certain restrictions, X(w) = x where w is in some original psp and x is in the support or range of the function X.

:As written, the CDF definition is back to being correct. [[User:Chjacamp|Chjacamp]] ([[User talk:Chjacamp|talk]]) 00:20, 24 November 2024 (UTC)

== Please clarify how area inequalities follow from diagram ==

The "Properties" section states the inequalities <math display="block">

x (1-F_X(x)) \leq \int_x^{\infty} t\,dF_X(t)

</math>

and

<math display="block">

x F_X(-x) \leq \int_{-\infty}^{-x} (-t)\,dF_X(t)

</math>

for a random variable <math>X</math> with a finite <math>L_1</math>-norm. These inequalities are straightforward to derive from the formula for the expected value of <math>X</math> using integration by parts. But the article claims that we can instead see why the inequalities are true from the included diagram by "consider[ing] the areas of the two red rectangles and their extensions to the right or left up to the graph of <math>F_X</math>". I don't see how these equalities follow from the diagram; please clarify. [[User:Ted.tem.parker|Ted.tem.parker]] ([[User talk:Ted.tem.parker|talk]]) 00:06, 24 March 2025 (UTC)

== Hatnote to pdf ==

There is [[Talk:Cumulative density function#RfC on redirect, disambig, article, or deletion|a discussion]] (RfC) on what to do with the page [[cumulative density function]]. One option is that it should redirect to this page. Some editors suggest that there should then be a hatnote here linking to [[probability density function]]. Please join the discussion! —[[User:St.nerol|St.Nerol]] ([[User talk:St.nerol|talk]], [[Special:Contributions/St.nerol|contribs]]) 16:59, 1 May 2025 (UTC)

[bitcoin-core-dev] 0.13 release schedule moved forward by a month [[User:Uvhw|Uvhw]] ([[User talk:Uvhw|talk]]) 02:57, 4 October 2021 (UTC)