Expected Value
\mathbb{E}_{\xi}[X]=q_0
}
Derivation
Eξ[X]=Eξ[i=0∑∞qiΦi(ξ)]=i=0∑∞qiEξ[Φi(ξ)]
Since Φ0=1 and all higher-order polynomials are constructed to have zero mean (i.e., E[Φi(ξ)]=0 for i>0 ), we get:
Eξ[X]=q0⋅1+i=1∑∞qi⋅0=q0
Variance
\mathbb{V}_{\xi}[X]=\sum_{i=1}^{\infty} q_i^2}
Derivation
Vξ[X]=Eξ[(X−Eξ[X])2]=Eξ(i=0∑∞qiΦi(ξ)−q0)2
Simplifying:
Vξ[X]=Eξ(i=1∑∞qiΦi(ξ))2=Eξ[i=1∑∞j=1∑∞qiqjΦi(ξ)Φj(ξ)]
Using orthonormality:
Vξ[X]=i=1∑∞j=1∑∞qiqjEξ[Φi(ξ)Φj(ξ)]=i=1∑∞j=1∑∞qiqjδij=i=1∑∞qi2
Assuming ξ is orthonormal.