In quantum mechanics the possible outcomes of a measurement of a quantity corresponding to a linear hermitian operator $A$ are only the eigenvalues $a_i$ of $A$ (see e.g. http://wwwitp.physik.tu-berlin.de/brandes/public_html/qm/umist_qm/node65.html). The expectation value of $A$ is computed as $$ \langle A \rangle_\psi = \sum_j a_j |\langle \psi | \phi_j \rangle|^2 $$ where $|\langle \psi | \phi_j \rangle|^2$ is called the transition probability (see https://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)).
In statistics a random variable $X$ is said to be a set of possible values $x_i$ from a random experiment (see e.g. https://www.mathsisfun.com/data/random-variables.html). The expected value of $X$ is computed as $$ \operatorname{E}[X] = \sum_{i=1}^\infty x_i\, p_i, $$ where $p_i$ are the probabilities of the $x_i$ (see https://en.wikipedia.org/wiki/Expected_value).
So can one say, that the set of expectation values $\{a_j\}$ with transition probabilities $|\langle \psi | \phi_j \rangle|^2$ in QM corresponds to a random variable $X = \{x_i\}$ with probabilities $p_i$ in statistics?
