PHY.8:4.3 - Derive the collective law without discarding dependence

Obtain the observable’s distribution by combining the admissible alternatives with their weights. For a classical state X with probability law P and an observable Y=g(X), the probability of a set B of outcomes is

P(Y in B) = integral 1[g(x) in B] P(dx).

The symbol 1 is one when its bracketed condition holds and zero otherwise. Use a weighted sum for a discrete state set. A non-ideal recording procedure adds its conditional response through MMP.7.

For a quantum state rho and an ideal measurement of observable A, outcome probabilities follow from the corresponding measurement projectors. When the second moment is finite, the mean is Tr(rho A) and the variance is Tr(rho A²) - Tr(rho A)². Here Tr denotes the trace, and A represents the physical observable under the selected measurement. When the required moments are not finite, use the outcome distribution or a relevant bounded event instead of these finite-moment summaries. Other measurement arrangements require their own operators and response.

Retain correlations while forming collective quantities. For a finite collection of jointly defined classical outcomes X_i with finite second moments,

E[sum_i X_i] = sum_i E[X_i],

Var(sum_i X_i) = sum_i Var(X_i) + 2 sum_(i<j) Cov(X_i,X_j).

Linearity of the mean needs no independence assumption. The variance equals the sum of individual variances when the total covariance contribution, 2 sum_(i<j) Cov(X_i,X_j), is zero. Pairwise zero covariance is sufficient, and independence is a stronger sufficient condition. Retain the dependence supplied by the physical arrangement.

A mean may be all that the question needs. When excursions matter, derive a relevant variance, tail probability or bound. Preserve units and the normalization used for comparison: the variance of a total and that of a per-constituent average differ by the square of the constituent count.