Expected Value

\mathbb{E}_{\xi}[X]=q_0 }

Derivation

Since and all higher-order polynomials are constructed to have zero mean (i.e., for ), we get:

Variance

\mathbb{V}_{\xi}[X]=\sum_{i=1}^{\infty} q_i^2}

Derivation

Simplifying:

Using orthonormality:

Assuming is orthonormal.