Quadrature to Approximate Expected Value

The general formula is an approximation based on a weighted sum of the model evaluated at specific points:

$$\mathbb{E}[\mathcal{M}(X)]={\int }_{{\Xi }_X}\mathcal{M}(x)f_X(x)\mathrm{d}x\approx \sum _{i=1}^{n_{\text{qu }}}{w}^{(i)}\mathcal{M}\left(x^{(i)}\right)$$

• $x^{(i)}$ → the integration points or nodes
• $w^{(i)}$ → the integration weights

• ${\int }_{{\Xi }_x}$ → represents integral over the support
• $\mathcal{M}$ → model function

Note:

• Polynomials of degree $2n_{qu}-1$ or less are integrated exactly.
• Points ${\left(x^{(i)}\right)}_{i=1}^{n_{qu}}$ are chosen as roots of polynomials ${\Phi }_i$ satisfying:

$${\int }_{{\Xi }_{\chi }}{\Phi }_i(x){\Phi }_j(x)f_{\chi }(x)\mathrm{d}x\propto {\delta }_{ij}$$