# Left “tail” of one-tailed distributions

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 $x\rightarrow a$?

• 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 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 8 '18 at 22:09
• @AdamO for a counterexample to the assertion "all real valued densities to go $0$," please see my post at 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. – whuber Feb 8 '18 at 23:44
• @whuber great examples of some really wonky distributions. Thanks for sharing. – AdamO Feb 8 '18 at 23:52