In statistics we have a concept called confidence region. This is rather uncomplicated for normal distributions, and possibly also for any other symmetric and/or monomodal distribution.
But what about finding $a,b$ (for density function $x\to p(x)$ of a one dimensional variable) : $$[a,b]\,\,\,\, s.t. \min_{\int_a^b p(x)dx = P}\{|a-b|\}$$
I would guess this interval shall be the same for monomodal distributions, but for multimodal distributions might give vastly different results. Could it still be useful? If so, how to calculate it?
Own work: Imagine we pick $P=25\%$ and Gaussian mixture with two gaussians of different $\mu,\sigma$ responsible for $50 \%$ total probability each. Now I imagine the above optimization should have minimum centered around the gaussian with lowest standard deviation. Is this correct or am I mistaken?
Imagine we call this first solution $(a_0,b_0)$, and then repeat this procedure but now we search $$a_1,b_1 \in \mathbb R \backslash ]a_0,b_0[$$ and after this we search $a_{k+1},b_{k+1}$ in $$\mathbb R \backslash \cup\{]a_0,b_0[,]a_1,b_1[,\cdots, ]a_k,b_k[\}$$
In our toy example Gaussian mixture if we successfully solve the optimization (and my intuition is correct), we would likely identify the two Gaussians by the intervals found $$]\mu_1-\xi_1,\mu_1+\xi_1[\,\,\,,\,\,\,]\mu_2-\xi_2,\mu_2+\xi_2[$$ And then decide $\sigma_1,\sigma_2$ from $\xi_1,\xi_2$