Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X_1,\dots,X_d\in L^\infty(\Omega,\mathcal F,\mathbb P)$ be bounded complex-valued random variables. Let $H:=L^2(\Omega,\mathcal F,\mathbb P)$, with [inner product](/page/Inner%20Product) \begin{align*} (f,g)_H:=\int_\Omega f(\omega)\overline{g(\omega)}\,d\mathbb P(\omega). \end{align*} For each $i\in\{1,\dots,d\}$, define the multiplication operator $M_i:H\to H$ by $M_i f:=X_i f$. Let $\Phi:\mathcal L(H)\to\mathbb C$ be the vector state \begin{align*} \Phi(T):=(T1_\Omega,1_\Omega)_H, \end{align*} where $1_\Omega$ is the constant-one element of $H$. Then $M_i^*=M_{\overline{X_i}}$ for every $i$, and for every integer $k\geq 1$, every choice of indices $i_1,\dots,i_k\in\{1,\dots,d\}$, and every choice of exponents $\varepsilon_1,\dots,\varepsilon_k\in\{1,*\}$, \begin{align*} \Phi(M_{i_1}^{\varepsilon_1}\cdots M_{i_k}^{\varepsilon_k})=\mathbb E[Y_1\cdots Y_k], \end{align*} where $Y_j=X_{i_j}$ if $\varepsilon_j=1$ and $Y_j=\overline{X_{i_j}}$ if $\varepsilon_j=*$.