Let $X$ be a random variable over the real line. Suppose that we know that $X$ is a Pearson distribution. Furthermore, suppose we know how the mass of $X$ is distributed into 6 intervals, so that if $F_X()$ is the cdf, we have $$F_X(-2) – F_X(\infty) = w_1$$ $$F_X(-1) – F_X(-2) = w_2$$ $$F_X(0) – F_X(-1) = w_3$$ $$F_X(1) – F_X(0) = w_4$$ $$F_X(2) – F_X(1) = w_5$$ $$F_X(\infty) – F_X(2) = w_6$$ Where $\sum_i w_i = 1$.
These mass weights are coming from analysts who are crudely estimating features of their prior distribution on the outcome of some something. For example you might ask an analyst for their estimates of where the S&P500 might be over the course of the next year, and they crudely assign probability to one of the 6 buckets. I'd like to map this in some principled way to a continuous distribution.
Is there an easy way to determine the maximum entropy probability distribution in the class of all Pearson distributions satisfying the above mass constraints?
Or, if this is too hard, is there a natural relaxation of my question that provides a "useful" distribution that meets these mass constraints?
Edited to explain motivation for the problem