Explicit formulae and implication of the expected values of some nonlinear statistics of tri-variate Gaussian variables

We obtain explicit formulae for the expected values $$E \{ \prod \nolimits _{i=1}^{3}g_i ( X_i ) \}$$ of standard tri-variate Gaussian random vector $${\underline{X}}= \left( X_1, X_2 , X_3 \right) $$ over the set $$g_i (x) \in \left\{ \delta (x), \mathrm {sgn}(x), |x|, x \right\} $$ of nonlinear and linear functions. Based on the results, we also suggest corrections to long-known formulae for two incomplete moments. Applications of the formulae in practical examples are also illustrated concisely. For easy reference for the readers, the explicit formulae derived in this paper and related well-known results are summarized in tables.