I am looking for a continuous probability density function defined in the interval [0, 1] with the following properties:

- If the mode is 0.5 then the mean is also 0.5
- If the mode is < 0.5 then: mean < mode
- If the mode is > 0.5 then: mean > mode
- The first derivative of the probability density function must be a continuous function.

I also wish to be able to have a shape parameter to control the distance between the mean and the mode

It would be even better if I could control the position of the mean relative to the mode via the parameter.

Does anyone knows such a function?

Thanks.