Uniquely determine the two parameters of a distribution given pre-determined probabilities for two disjoint subsets of its support

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?

• sorry, i slightly abused the word 'disjoint' as clearly in example one the subsets $(-\infty, 0]$ and $[0, 1]$ are not disjoint. Please pardon the sloppiness in the definition here.
– Alex
Commented Jan 18, 2016 at 3:24
• Have you ever seen (or sketched for yourself) a normal density curve? If you have, and have paid any attention to the details, the datum $F(0)=0.5$ is wiggling its eyebrows and winking an eye, and pointing out with suggestive movements of its elbow and shoulder and thumb, the value of the mean to you. Commented Jan 18, 2016 at 7:39
• Oh yes, you are right, that is a bad example, I will change it.
– Alex
Commented Jan 18, 2016 at 8:15
• This note is relevant: johndcook.com/quantiles_parameters.pdf . It shows how to solve this problem for the normal, log normal, Cauchy, Weibull, gamma, and inverse gamma probability distributions.
– Alex
Commented Jan 18, 2016 at 22:43
• thanks @Glen_b, fixed now. Anyway, by the linked paper, I suppose the answer to 2) is also yes, since the skew normal distribution forms a location scale family.
– Alex
Commented Jan 19, 2016 at 4:25

For your first example, $$F(0) = \Phi\left(\frac{0-\mu}{\sigma}\right) = \Phi\left(\frac{-\mu}{\sigma}\right) = 0.4$$ and so $$-\frac{\mu}{\sigma}= \Phi^{-1}(0.4) \approx -0.2534\tag{1}$$ while $$F(1) = 0.7 = \Phi\left(\frac{1-\mu}{\sigma}\right)$$ and so $$\frac{1-\mu}{\sigma}= \Phi^{-1}(0.7) \approx 0.5243\tag{2}$$ and you should be able to calculate the values of $\mu$ and $\sigma$ from $(1)$ and $(2)$.