Let $F(x| \alpha, \beta)$ denote the cumulative density function of a probability distribution. Let $[a,b]$ and $[c,d]$ be two disjoint subsets of the support of $F$. Suppose that $F(b) - F(a) = p$ and $F(d) - F(c) = q$, for some non zero $p$ and $q$ such that $p + q < 1$. Can you always uniquely determine $\alpha$ and $\beta$? Otherwise, under what conditions/probability distributions can you determine the parameters?
For examples:
1) $F$ is the normal distribution, and $F(0) = 0.4$ and $F(1) - F(0) = 0.3$. Can you uniquely determine the mean and standard deviation?
2) Consider the skew normal distribution, with probability density function $f(x| \xi, \omega, \alpha) = 2\phi(\frac{x-\xi}{\omega})\Phi(\alpha \frac{x-\xi}{\omega})$. If we fix the location $\xi$, can we uniquely determine the scale $\omega$ and skew $\alpha$ under conditions set out above?