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domain and support are not the same thing.

Per mathworld

The term domain is most commonly used to describe the set of values for which a function (map, transformation, etc.) is defined. For example, a function f(x) that is defined for real values $x \in R$ has domain $\mathbb R$, and is sometimes said to be "a function over the reals."

Per wiki

the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

Consider the normal distribution, it seems that the domain is equal to the support, is it reasonable to say this? In another word, if I ask some mathematicians if the domain of the normal distribution is equal to its support, what would they say? Yes? or, They are not comparable?

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The normal distribution is a distribution, not a function, so neither of the definitions you have given apply to it.

What is a distribution? There are several subtly different ways they are defined rigorously, not all of which are well-represented on wikipedia. But none of them coincides with the notion of a function. In its simplest form, a distribution is a mathematical object that assigns to each interval a number, satisfying some natural conditions. In this situation, the support is the set of points with the property that every non-empty open interval containing the point is assigned a strictly positive number.

If the distribution happens to have a density (which is a function) then the support of the distribution is essentially the same thing as the set of points on which the density function does not vanish.

Note, I purposely did not mention "domain" above since I do not know of a precise definition that applies to distributions. Maybe you can specify one if you have it in mind?

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  • $\begingroup$ I considered getting pedantic, but I assumed by 'normal distribution' the user is referring to the probability density function of the standard normal distribution. $\endgroup$
    – Joe
    Aug 9, 2019 at 3:03
  • $\begingroup$ I appreciate your sentiment, but the point of the question is pedantic by nature so I feel a fully correct (and therefore pedantic) answer is important to resolve the confusion. I believe this user has been asking a few similar questions recently, involving the confusion between random variables, their domains, and the support of their laws. $\endgroup$
    – pre-kidney
    Aug 9, 2019 at 3:04
  • $\begingroup$ Fair enough. I'm never sure how detailed to get to clear up confusion without adding to it. $\endgroup$
    – Joe
    Aug 9, 2019 at 3:06
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    $\begingroup$ This is always an issue and is completely dependent on context, which can be hard to get on an online forum like this one, especially considering that many people from all sorts of backgrounds may look at any given answer. In such a situation I think it is better to avoid spreading misinformation at the cost of being pedantic - after all, it is a math.SE and not applied math / stats / physics / etc where other answers that are more useful to practitioners might be appropriate. $\endgroup$
    – pre-kidney
    Aug 9, 2019 at 3:10
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    $\begingroup$ It is both a generalized function, and a probability distribution (but not if you multiply it by 2 - that was for illustrative purposes). A measure is a set function, but it is misleading to call it a function in this context since the question is implicitly referring to functions mapping real numbers to real numbers, and not functions mapping sets of real numbers to sets of real numbers. They are very different types of objects in this context... $\endgroup$
    – pre-kidney
    Aug 9, 2019 at 4:50
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Technically, I believe the support is the smallest closed subset of the domain containing all points that are not mapped to zero.

But yes, both the domain and the support for the normal distribution are all real numbers.

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  • $\begingroup$ The normal distribution is not a function, so neither definition given in the question applies to it. $\endgroup$
    – pre-kidney
    Aug 9, 2019 at 2:53

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