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A list of discrete event distributions are labeled as either with or without finite support.

https://en.wikipedia.org/wiki/List_of_probability_distributions#Discrete_distributions

What does it mean for a distribution to have finite support? It's been a long time since I have sat in a statistics classroom, so it would be appreciated if the explanation started with ELI5 and graduated into a more nuanced explanation.

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    $\begingroup$ See the first sentence here for what 'support' means in mathematics (the subset of the domain containing those elements which are not mapped to zero); that's best to have clear first. Speaking loosely finite support means there's a finite number of elements in that subset. (More formally, the definition in probability is given later in the article; for my informal description to work we must be using the counting measure.) $\endgroup$ – Glen_b Sep 27 '18 at 16:31
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A discrete distribution with finite support can only have a finite number of possible realizations. For example, the distribution associated with a coin toss experiment is discrete with finite support as you can only observe one of two values: 'Heads' or 'Tails'.

A discrete distribution with infinite support can have an infinite number of possible realizations. For example, the distribution associated with the number of customers who arrive at a bank between 9 am - 9:30 am is discrete with infinite support as you can potentially observe many values: 0, 1, 2, 3, ...

Note: In the second paragraph, we ignore the reality of a finite population on Earth.

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Discrete distribution most frequently refers to a distribution that only achieves integer values, and more generally a discrete distribution defines probabilities for distinct potential outcomes that do not cover any continuous interval range. For example, a Poisson distribution often is used to model the number of radioactive decays detected, and those can be any natural whole number including zero counts.

Finite support refers to the range of values that a distribution can achieve. For example, two dice can have an summed outcome from 2 to 12 of the number of dots face up on those thrown dice. We can symbolize this as some integer value $i$ on the interval $[2,12]$. Those dice outcomes are also discrete outcomes. Thus, the distribution of outcomes is discrete with finite support.

Armed with this information, take a look again at the Wikipedia entry. That entry also details other distributions and other support. For example, other distributions can have semi-infinite support, e.g., on $[0,\infty)$ and be discrete (Poisson) or or continuous (Gamma distribution).

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    $\begingroup$ I'm uncomfortable with the phrasing in the opening sentence. I don't think there's any misunderstanding on your part but I worry that it will be somewhat likely to convey a misunderstanding. Certainly the most common applications involve integers, but I am concerned that the first sentence shouldn't give the impression that "discrete" means "integer". The use of the word "means" is my particular concern in that first sentence. $\endgroup$ – Glen_b Sep 27 '18 at 16:27
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    $\begingroup$ @Glen_b Yeah, me too. So I changed it. $\endgroup$ – Carl Sep 28 '18 at 6:33

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