A two-state uncertain reaction rate has an exact degree-one chaos expansion

The first two moments are $\mathbb{E}[Y]=0$ and $\mathbb{E}[Y^2]=1$, so the degree-one Gram matrix is $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, positive definite. Thus [F1] applies. Gram--Schmidt gives ${\psi }_0=1$ and ${\psi }_1=Y$, since $Y$ already has mean zero and unit norm and its leading coefficient is positive.

Substitution gives $k(Y)=\kappa {\psi }_0+\kappa \eta {\psi }_1$ at both support points, so the degree-one projection residual is identically zero. Its coefficients are $c_0=\mathbb{E}[k(Y)]=\kappa$ and $c_1=\mathbb{E}[k(Y)Y]=\kappa \eta$. By [F3], the mean is $c_0$ and variance is $c_1^2={\kappa }^2{\eta }^2$.

The condition $\mathrm{\mid }\eta \mathrm{\mid }<1$ implies $1+\eta Y\ge 1-\mathrm{\mid }\eta \mathrm{\mid }>0$ at both states, so the reaction rate is positive on its whole support. Every function on this support also has the exact form $g(Y)=\frac{g(1)+g(-1)}{2}+\frac{g(1)-g(-1)}{2}Y$; hence this two-dimensional finite basis spans $L^2(\rho )$ for this particular law.