Why is instead of the standard sandwich integral?
The sandwich integral is for quantum mechanical observables (where the operator is the filling and the wavefunction and its complex conjugate are the bread). Here, we are just talking about an ordinary classical expectation value: this expression is just the mean value of something (like for the probability/error analysis stuff from the beginning of semester). It will be the sum of energies, weighted by probability of getting that energy, and normalized by total probability.
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