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There is one accepted answer, but I don't get it quite. In fact that answer raised the few additional questions I liked to share.

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1. How would you define the tails of the distribution?

Seams that this is a fundamental question. In this question you used the mathematical formulation of survival function but I guess you may define the tails based on areas: $(-inf, μ−3σ]$ and $[μ+3σ,inf)$.

My hypothesis is if some distribution has "fatter tail" that would be larger area under the PDF curve in that tail area. Is this a good definition of tails of just a simplification?

I can agree that on Wikipedia at the moment there are several suspicious satements at least regarding the fat tailed distribution evaluation based on skewness and kurtosis examination.

You confirmed: Please note that moments, like skewness or kurtosis, are not good for characterizing tail behavior.

2. Why the density is not the correct object to study?

PDF for continuous functions seams like the best possible function to study because it must add to 1 by definition. So far I meat some distributions like Cauchy that don't have Moment Generating function i.e. don't have moments but how can continuous distribution be without a PDF?

Isn't that if we know the CDF ( I like the term cumulative in here) we should know the PDF?

3. CDF should be continuous, and this is the reason for the 2.

This is more like helping me to understand the dynamics behind your answer.

4. The terms fat-tailed and heavy-tailed are sometimes synonymous

This is from Wikipedia. Would you agree these are synonyms.

In here I think the term "long-tailed" is definitely not the synonym with the upper two. Possible confirmation would be fine.

Easy Points
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  • Four questions is too many to pose at once. The fourth is answered at https://stats.stackexchange.com/questions/168851, so I will mark that as a duplicate. The first three are predicated on the incorrect assumption that all random variables have PDFs--they don't. Generally, when you have focused questions concerning a particular post here, please use the commenting facility to inquire first: you have the reputation to do that. – whuber Feb 25 '21 at 18:08
  • [You got](https://stats.stackexchange.com/questions/86429/which-has-the-heavier-tail-lognormal-or-gamma) concrete question for the distributions that have PDF and you used the assumption that "some distributions" don't have the PDF. This is how complicated answers are born. :) The question you directed me is my only satisfaction. – Easy Points Feb 25 '21 at 18:27
  • You have it backwards. "Some distributions don't have a PDF" is not an assumption, it's a fact. And it's not just a mathematical nicety: many of the commonest and most useful distributions do not have PDFs. When a concept (like fat, heavy, or long tails) applies to all distributions it is often preferable to explain it in a way that does not require special assumptions like "has a PDF." – whuber Feb 25 '21 at 18:54
  • @whuber, I am balancing and measuring not to ask super evident and overly simplistic questions, at least we are working with non discrete = PDFs. What I learned so far the most common examples of continues distributions are gamma (exponentiation), normal, uniform, beta distribution as distribution of distributions. I am limited, so I searched last 30 minutes to find continuous distributions that don't have PDF in explicit form. So what you just stated is *new* to me. Can you share a link so I can read more about these continuous distributions without PDF. – Easy Points Feb 25 '21 at 19:21

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