Start with $\text{global minimum}=0, \text{target}=:\text{mean}=\bar{x}, \text{global maximum}→\text{infinity}$ corresponding to a smooth continuous probability distribution of $x$.
Now suppose that the global min and max are substantially lower and higher than values likely to occur. Suppose each <0.03% likliehood; then 99.94% probability that $x$ will occur between $a$ and $b$ (where $0<a<x<b<∞$, then $P(a<x<b)=0.9994$), and 100% probability that x will between 0 and infinity (that is, $P(0<x<∞)=1$). Then what form of equation can describe the distribution of $x$ about the target?
Now suppose one modification: the target is closer to $a$ than $b$ (that is, $a-x > b-x$). Is this possible (consistent with the preceding parameters)? If so, what form would the equation take? And what modification could lower the global minimum to a nonzero value (xeither negative or positive $<a$) and likewise to lower the global maximum to a number (smaller than infinity but greater than $b$), or alternatively to make the global min become $a$ andor global max to become $b$ (i.e. $P(a<x<b)=1$, where discrete approximation would have $P(x=a)=P(x=b)=0$)?
The general idea I'm after is to have $P(\bar{x}<x)>P(x<\bar{x})$ where $x>0$ (and $0<P(\cdot)<1$), and understand how to modify said model from there.
A distribution that is strictly positive and skewed right can be modeled by a log-normal distribution. https://en.wikipedia.org/wiki/Log-normal_distribution