Seeking "Flexible" Probability Distribution with Compact Support?

Question

I am looking for a probability distribution that

1. is continuous with compact support $[a,b]\subset \mathbb{R}$
2. is "flexible" in that its mean, variance, and skew can be tractably manipulated (e.g. by changing parameters).

To illustrate what I mean, here are some examples of distributions that partially accomplish the above:

• the raised cosine distribution allows for flexible manipulation of its variance.
• the wrapped normal and von Mises distributions allow one to flexibly manipulate the mean and variance.

Does anyone know of a continuous distribution with compact support whose mean, variance, and skew can be systematically manipulated?

Apologies for the imprecise question. Please let me know if you would like me to clarify anything.

Answer 1

The beta distribution is commonly used as a "swiss army knife" of probability distributions with finite support. If the support is fixed, you only have two parameters, so your mean, variance, and skew may not be independently selectable .