Generate a random distribution. How to generate the distribution itself?

Does anyone know how to generate a random distribution (i.e. a distribution with random appearance -- like 'letting a child draw a function graph at random'-random) with positive mean value less than 1?

I want to define a set of probability distributions that all are subject to the constraint 'the mean is in (0,1)'. The distributions should be 'random selections from the space of probability distributions'. It's okay if that's a bit unclear. Any answer that is creative is welcome!

Solutions in Python? Please post it :)