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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">
== Definition as expectation value ==
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>
''( where '''1'''<sub>{''X ≤ x''}</sub> 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'''''
: <math> \varphi_X(t) = \operatorname{E} \left [ e^{itX} \right ]</math>
''also completely determines behavior and properties of the probability distribution of the random variable ''X''. The two approaches are equivalent in the sense that knowing one of the functions it is always possible to find the other, yet they both provide different insight for understanding the features of the random variable.''
The notation <math>F_X(x) = \operatorname{E} \left [\mathbf{1}_{\{X\leq x\}} \right]</math> is so confusing I ask the community a clarification in this page ([[Characteristic function (probability theory)|Characteristic function]]).
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