PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.CStarAlgebra.Statistics

Let $A$ be a star-algebra and $a \in \text{selfAdjoint}(A)$ be a self-adjoint element. For any natural number $n \in \mathbb{N}$, the $n$-th Jordan power of $a$, denoted $a^{[n]}$ and defined recursively by $a^{[0]} = 1$ and $a^{[n+1]} = a \circ a^{[n]}$ (where $\circ$ is the Jordan product), is equal to the ordinary power $a^n$ in $A$.

Let $A$ be a unital $C^*$-algebra and $\omega$ be a state on $A$ (a unital positive linear map from $A$ to $\mathbb{C}$). For any self-adjoint element $a \in A$ (an observable) and any natural number $n \in \mathbb{N}$, the $n$-th moment of $a$ in the state $\omega$ is given by the integral of the $n$-th power function with respect to the real spectral measure $\mu_{\omega, a}$ associated with $\omega$ and $a$: $$\text{moment}_n(\omega, a) = \int_{\mathbb{R}} y^n \, d\mu_{\omega, a}(y)$$ Here, the moment is defined by applying the state to the $n$-th Jordan power of the observable, which for a single observable coincides with the ordinary power $a^n$.

Let $A$ be a $C^*$-algebra, $\omega$ be a state on $A$, and $a$ be a self-adjoint element (an observable). The value of the state $\omega$ on the observable $a$ is equal to the integral of the identity function with respect to the real spectral measure $\mu_{\omega, a}$ associated with $\omega$ and $a$: $$\omega(a) = \int y \, d\mu_{\omega, a}(y)$$ This identifies the mean (the expected value) of the observable as its first moment.

Let $A$ be a unital $C^*$-algebra, $\omega$ be a state on $A$, and $a \in A$ be a self-adjoint element (an observable). Let $\mu_{\omega, a}$ denote the real spectral measure (the probability distribution) of $a$ in the state $\omega$. The variance of $a$ with respect to $\omega$ is equal to the difference between the second moment and the square of the first moment of its spectral measure: $$\text{Var}_{\omega}(a) = \int_{\mathbb{R}} y^2 \, d\mu_{\omega, a}(y) - \left( \int_{\mathbb{R}} y \, d\mu_{\omega, a}(y) \right)^2$$