Example of left-tail distribution Assuming that $X>0$, a lognormal distribution for such a random variable is considered right-tailed because it skews towards the right. 
Out of curiosity, is there a well-known PDF that is left-tailed, i.e., skews to the left for $X>0$? 
Thanks.
 A: In general, distributions with tails are defined base on the definitions of the random variable in question. For example, for a random variable $X$ representing the amount of rainfall or revenue or population income. These all take only positive values such that $X \ge 0$ and they tend to follow very right-tailed distributions, such as Log-Normal, Gamma, Weibull and other common distributions. 
However there are much fewer examples in common literature for random variables that are left-tailed and take values that skew to the left. There is a wide variety of data that exhibits this property, such as survival of human life (age of mortality) or scores of easy tests or simply flipping a weighted coin. It is generally tricky to parameterise a distribution which has a cutoff $<\infty$ so we don't really see too many formal well-known distributions that satisfy the left-skew. For another random variable, say $Y \ge 0$, as mentioned in the comment above, the Beta , Binomial, Dirichlet and even a Generalised Extreme Value distributions can be left tailed.


In fact for some random variable $Z \in \Bbb R$ there are negatively-skewed distributions without placing a restriction on the strictly positive conditions used in economics for historical analysis of efficiency and by big banks and financial institutions for expected profits and regulatory requirements (skewed Normal).

