I think of the "tail" of a probability distribution as the behavior of its PDF $f(x)$ as $x\rightarrow +\infty$. For some PDFs with complicated expressions, it is sometimes easy to study their limiting behavior ($x\rightarrow +\infty$), or equivalently their tail, because it compares to that of standard distributions. In the case of a one-tailed distribution supported on $[a,+\infty)$, is a concept of a "right tail" defined for when $x\rightarrow a$?
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Your "tailedness" understanding is a misstep. All real valued densities go to 0. The only semi-related concept I'm aware of is being [bounded in probability](https://math.stackexchange.com/questions/929319/does-bounded-in-probability-imply-convergence-in-probability) for something like a sample mean. Tails are in general not well defined, but at the very least could be the 0-49th quantiles for the left tail and 51-100th quantiles for the right tail. – AdamO Feb 08 '18 at 22:09
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1@AdamO for a counterexample to the assertion "all real valued densities to go $0$," please see my post at https://stats.stackexchange.com/a/86503/919. In the sense of asymptotic behavior of the distribution function $F$, tails are indeed well-defined. There are always two of them, because (by definition) all distribution functions are defined on $\mathbb{R}$, [which has two ends](https://en.wikipedia.org/wiki/End_(topology)). – whuber Feb 08 '18 at 23:44
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1@whuber great examples of some really wonky distributions. Thanks for sharing. – AdamO Feb 08 '18 at 23:52