Table 4.

Comparison of integration techniques, polynomial quadrature and with integration for different random distribution.

Distribution/polynomial	Quadrature	$⟨{\mathrm{\Phi }}_{\mathrm{\alpha }}(\xi ){\mathrm{\Phi }}_{\mathrm{\beta }}(\mathrm{\xi })⟩={\mathrm{\delta }}_{\mathrm{\alpha }\mathrm{\beta }}{\mathrm{\gamma }}_{\mathrm{\alpha }}$	n = integration points
Uniform/Legendre	Clenshaw–Curtis29–31	${\gamma }_{\alpha }=\frac{2}{2\alpha +1}$	2N+1
Normal/Hermit	Gussi-Hermit	${\gamma }_{\alpha }=\sqrt{\pi }{2}^{\alpha }\alpha !$	N + 1
Truncated normal/Hermit	Gussi-Hermit	${\gamma }_{\alpha }=\sqrt{\pi }{2}^{\alpha }\alpha !$	N + 1