Looking for discrete skewed gaussian(-like) probability distribution function

Looking for discrete skewed gaussian(-like) probability distribution function

I'm faced with a modelling task in which I have a frequency that can vary
in a certain range. The variation is discrete and resembles a (skewed)
gaussian distribution. What I need is a function that takes an input value
(the nominal frequency value) x and a uniform distributed value u=[0,
1.0], two parameters ([a, b]) which represent the range of the variation
and returns y (y will be within this range).
For instance, take a value of 100Hz as x and a range of [40, 110], I'd
like to obtain a random y that lies in the range with a median of 100.
Is there a suitable probability distribution function that I could use?
I've looked at the binomial distribution, but thought it's unsuitable, as
it doesn't have a skew parameter. Any help is appreciated.