MMP.17:5.3 - Reuse stochastic cases when the requested response changes

A stochastic loss model returns either (0) or (5). Its source structure states that the probability (p(x)) of loss (5) is affine for (0\leq x\leq1); the two endpoint probabilities are unknown. All 2,000 runs in the construction are independent: 1,000 runs give 100 losses of (5) at (x=0), and another 1,000 give 300 at (x=1).

Fit the endpoint proportions and interpolate:

[ \widehat p(x)=0.1+0.2x,\qquad \widehat\mu(x)=5\widehat p(x)=0.5+x. ]

The affine premise comes from the source structure, not from the two observed proportions. The mean surrogate is evaluable without rerunning the stochastic source.

At (x=0.5), the estimated mean is (1). A simple uncertainty calculation illustrates what must accompany that value. Each endpoint proportion has variance at most (1/(4{,}000)). Independence gives variance at most (1/(8{,}000)) for their average, an unbiased estimator of (p(0.5)) under the affine premise. Chebyshev’s inequality therefore gives coverage of at least 95% for a half-width of (0.05) around that average. The resulting probability interval is ([0.15,0.25]), and the corresponding mean interval is ([0.75,1.25]). For a receiver using this 95% confidence procedure, the upper endpoint supports the mean-at-most-(1.4) comparison; it is not a deterministic bound. MMP.13 permits sharper uncertainty calculations when the receiving use needs them.

The receiver now asks whether (\Pr(Y_{0.5}>4)\leq0.18). The mean alone cannot answer: a constant loss of (1) has the same mean but a different tail. Here the retained two-point support supplies the relation (\Pr(Y_x>4)=p(x)=\mu(x)/5). Reuse the same cases and the probability surrogate; no new fit is needed. The interval ([0.15,0.25]) crosses (0.18), so this uncertainty result leaves the new comparison unresolved.

The interval concerns the fixed input (0.5) under independent runs, the stated support and the affine probability law. It is not a simultaneous guarantee for all inputs, and it does not cover error in those source premises. A sharper inference, a useful bound, more runs or a qualified unresolved answer are different possible continuations; choose among them for the receiving question.