An observable in quantum mechanics is a measurable quantity in an experiment or observation. A postulate for building the mathematical theory of quantum mechanics is that
2.With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction.
The postulates can be found in the second page in this link.
These postulates are in the foundations of building up mathematical theories of quantum mechanics , all of them, from Hilbert spaces to quantum field theory.
It is simpler when talking of probabilities of getting a value of an observable to discuss it in the original wave function framework of probabilities. A single measurements picks up an instant from the predicted by the theory probability distribution. This is the same as with the statistical mechanics probability distributions.
Where the two diverge are on the mathematics behind them. For random variables one has Gaussian, or Poisson or ...distributions that are an average over dynamical functions derivable from classical mechanics, for example, with a few assumptions on the way the sample under study behaves.
For quantum mechanics the probabilities are the dynamical functions determined from first principles. There exist no further internal dynamics that generate the probability distributions.
Thus statistical probabilities with assumptions of randomness cannot be equivalent with probabilities determined dynamically.
The set of realizations of a random variable would then be the spectrum (set of eigenvalues) of an operator.
One would have to define the mathematical space on which this operator would work, giving the continuum you call "spectrum".
So could I tell a statistician that a "random variable" is in fact an operator?
Statistical mechanics is a meta level on classical mechanics. Quantum mechanics is the first level , no underlying structure deterministic or not, has been discovered.
In addition, it is not the variable that is an operator, the value of the variable instance of a variable, measurment, is an entry in the probability distribution .
You could say an "operator will give me the measurement of temperature in this sample", but there is no mathematical theory of operators that will act on the collective sample , for example a "temperature operator" to give the probable temperature on a specific thermometer measurement.
Probabilities in quantum mechanics are predictive because they are not uniform, giving discrete spectra.
So the analogy does not hold.