Are there any well-known examples (preferably from the exponential family) for left skewed probability distributions (i.e. negative skewness) on either $[0, \infty)$ or $\mathbb{N}_0$ ?
The only left-skewed distributions I could find are the beta distribution on bounded intervals for $\beta > \alpha$ and the binomial distribution on $\{1,2,...n\}$ for $p>1/2$.