$\left \vert\varepsilon _K \right \vert$

Since the third constraint depends on two strongly correlated parameters, we need to consider a joint bivariate Gaussian distribution:

$$f(c,d) = \frac{1}{2\,\pi\,\sigma_c\,\sigma_d\,\sqrt{1-\rho_{cd}^2}}\cdot \... ...rac{1}{2\,(1-\rho_{cd}^2)} \left[ \frac{(x-\mu_c)^2}{\sigma_c^2} \right.\right.$$

$$\left.\left. - 2\,\rho_{cd}\,\frac{(c-\mu_c)(d-\mu_d)}{\sigma_c\,\sigma_d} + \frac{(d-\mu_d)^2}{\sigma_d^2} \right] \right\} \,,$$ (14)

with $\mu_c=\mbox{E}[c]=3.4$, $\sigma_c=0.9$, $\mu_d=\mbox{E}[d]=1.23$, $\sigma_d=0.33$, and $\rho_{cd}=\rho(c,d)= 0.76$. The integral

$$f(\bar {\rho},\bar{\eta}\,\vert\,C_3) = \int\!\!\!\!\int_{-\... ...+c\,(1-\bar {\rho})]-d\right) \, f(c,d)\,\mbox{d}c\,\mbox{d}c$$ (15)

can be still evaluated analytically but we omit here the final formula, just giving the 3-D plot of the resulting p.d.f. normalized in the region of interest (Fig. 5).

Figure: Probability density function of $\bar {\rho}$ and $\bar{\eta}$ obtained by the constraint given by $\left \vert\varepsilon _K \right \vert$.