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This statement "the probability that a continuous variable exactly equals to a given value is zero" is commonly used.

is this an axiom, a property of continuous variable, a rule of integration or a conclusion of measure theory?

please provide a citation of a textbook or formally publication.

the wiki itself some answer linked is being tagged "unclear sources"

Arthur
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JJJohn
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  • @JMoravitz please provide a citation of a textbook or formally publication.

    the wiki itself your answer linked is being tagged "unclear sources"

     – JJJohn Nov 14 '19 at 15:10
  • @JMoravitz I am not asking for an explanation. I am asking for a solid sources. – JJJohn Nov 14 '19 at 15:12
  • Why are you shouting? – saulspatz Nov 14 '19 at 15:12
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    It is a property of a continuous random variable. If it is true for every value then the random variable is by definition a continuous random variable. – drhab Nov 14 '19 at 15:12
  • It can be proven by anybody. It doesn't need a clear original source since it is such a basic application of fundamentals. Do we need a cited source when talking about the color of the sky? If you want to cite something, then cite the linked answer. – JMoravitz Nov 14 '19 at 15:13
  • @saulspatz because someone is trying to close my question without considering what I am actually asking. – JJJohn Nov 14 '19 at 15:14
  • Someone has voted to close your question as a duplicate. If you think that the answers given to that question don't answer yours, please explain why not. How is your question different? You should this in the body of the question, not the comments. – saulspatz Nov 14 '19 at 15:18
  • @saulspatz I've done that, although obviously JMoravitz doesn't buy that. – JJJohn Nov 14 '19 at 15:26
  • Any decent textbook of probability theory should contain a definition of "continuous random variable" (if it doesn't, I wouldn't call it decent). – Robert Israel Nov 14 '19 at 15:28

2 Answers2

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A random variable $X$ is by definition a continuous random variable if and only if $P(X=x)=0$ for every $x\in\mathbb R$.

drhab
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  • If you define a continuous r.v. so, then you need to define absolutely continuous r.v., and then it turns out that absolutely continuous r.v. is all that the probability theory really needs; I prefer to reserve the term continuous r.v. for a random variable having probability density. – kludg Nov 14 '19 at 16:10
  • @kludg Well, so be it. For me nothing is wrong with the term "absolutely continuous" for a random variable that has a probability density. IMHV it is narrow-minded to say that then it turns out that absolutely continuous is all that probability theory really needs. – drhab Nov 14 '19 at 16:31
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This washington handout gives

Property 1.1. If X is a continuous rrv, then For all $a\in\mathbb R, \ \mathbb P(X = a) = 0$

drhab
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JJJohn
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