  The beta distribution is the best-known distribution for modelling
doubly-bounded data, \eg percentage data or probabilities. A new generalization
of the beta distribution is proposed, which uses a cubic transformation of the
beta random variable. The new distribution is label-invariant like the beta
distribution and has rational expressions for the moments. This facilitates its
use in mean regression. The properties are discussed, and two examples of
fitting to data are given. A modification is also explored in which the
Jacobian of the transformation is omitted. This gives rise to messier
expressions for the moments but better modal behaviour. In addition, the
Jacobian alone gives rise to a general quadratic distribution that is of
interest. The new distributions allow good fitting of unimodal data that fit
poorly to the beta distribution, and could also be useful as prior
distributions.
