# The algebra of interpolatory cubature formulæ for generic nodes

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## Abstract

We consider the classical problem of computing the expected value of a real function *f* of the *d*-variate random variable *X* using cubature formulæ. We use in synergy tools from Commutative Algebra for cubature rulæ, from elementary orthogonal polynomial theory and from Probability.

## Keywords

Design of experiments Cubature formulæ Algebraic statistics Orthogonal polynomials Evaluation of expectations## Notes

### Acknowledgements

G. Pistone is supported by de Castro Statistics Initiative, Collegio Carlo Alberto, Moncalieri Italy. E. Riccomagno worked on this paper while visiting the Department of Statistics, University of Warwick, and the Faculty of Statistics at TU-Dortmund on a DAAD grant. Financial support is gratefully acknowledged. The authors thank Prof. H.P. Wynn, Prof. G. Monegato (Politecnico di Torino) and Prof. Dr. Hans Michael Möller (Technische Universität—Dortmund) for their useful suggestions.

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