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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?

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  • $\begingroup$ 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. $\endgroup$
    – Alex
    Commented Jan 18, 2016 at 3:24
  • $\begingroup$ 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. $\endgroup$ Commented Jan 18, 2016 at 7:39
  • $\begingroup$ Oh yes, you are right, that is a bad example, I will change it. $\endgroup$
    – Alex
    Commented Jan 18, 2016 at 8:15
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    $\begingroup$ 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. $\endgroup$
    – Alex
    Commented Jan 18, 2016 at 22:43
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    $\begingroup$ 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. $\endgroup$
    – Alex
    Commented Jan 19, 2016 at 4:25

1 Answer 1

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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 $(2)$.

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  • $\begingroup$ yes, this forms a system of linear equations and the initial conditions forces it to be a consistent system. $\endgroup$
    – Alex
    Commented Jan 19, 2016 at 22:09

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