I am looking for a parameterized family of probability densities on the unit interval, which all share a strictly positive density at zero. Ideally, they could be parameterized by some parameter $\lambda$, such that a higher $\lambda$ implies first order stochastic dominance. Is there something like this, or maybe close to it?
Thanks!
You could form a density like this using a mixture of Beta distributions. For example, we could take a mixture of a uniform distribution (to give non-zero density at all points) and another Beta distribution parameterised with $\lambda$ as the mean. This gives the general form:
$$f_X(x) = \phi + (1-\phi) \cdot \frac{x^{\lambda \kappa-1} (1-x)^{(1-\lambda) \kappa -1}}{B(\lambda \kappa, (1-\lambda) \kappa)} \quad \quad \quad \text{for all } 0 \leqslant x \leqslant 1.$$
where $0 < \phi < 1$ and $\kappa >0$. This distribution has non-zero density at all points and higher values of $\lambda$ give higher values (in the sense of first-order stochastic dominance). In particular, it has mean:
$$\mathbb{E}(X) = \frac{\phi}{2} + (1-\phi) \lambda.$$