Exact nonlinear uncertainty example
In a constructed scalar model Y=X² with X uniformly distributed on [−a,a], a>0, exact distribution propagation gives E(Y)=a²/3 and Var(Y)=4a⁴/45, hence u(Y)=2a²/√45.
Key knowledge
In a constructed scalar model Y=X² with X uniformly distributed on [−a,a], a>0, exact distribution propagation gives E(Y)=a²/3 and Var(Y)=4a⁴/45, hence u(Y)=2a²/√45. Linearization at the input mean zero instead gives f(0)=0 and f′(0)²Var(X)=0; it misses both the positive mean and the nonzero spread.
Values and equations
X∼Uniform(−a,a); Y=X²; E(Y)=a²/3; Var(Y)=a⁴/5−a⁴/9=4a⁴/45; u(Y)=2a²/√45
- X,a
- Uncertain scalar and fixed positive half-width; same input unit
- Y,E(Y),u(Y)
- Output, its estimate and standard uncertainty; squared input unit
- Var(Y)
- Output variance; fourth power of input unit
Conditions and scope
- Illustrative analytic model with the stated uniform input knowledge, a fixed exact bound and output unit equal to the square of the input unit; it is not an experimentally certified measurement result.
- Integrating X² and X⁴ yields a²/3 and a⁴/5; the asymmetric output distribution must be retained for coverage. A zero local derivative is not evidence of zero uncertainty. Use appropriate higher-order or distribution propagation when linearization is inadequate; Gaussian-specific moment corrections are not assumed for this uniform example.
Sources · 2
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Bureau International des Poids et Mesures (BIPM)2 sources
Evidence and review
JCGM 101:2008: propagation of distributions: §5.2–§5.4, printed pages 8–9: distribution mapping, expectation/standard-deviation summary and analytic implementation; §6.4.2: rectangular input density. The X² example is explicitly constructed here: integrate (1/(2a))x² and x⁴ over [−a,a], subtract squared mean, rather than claiming the document contains this exact example. JCGM 100:2008/Amd.1:2026: nonlinearity in measurement models: Printed page 1: new note 1 to 4.1.4 requires appropriate nonlinear treatment for the estimate as well as uncertainty; higher-order formula stated there assumes independent normal inputs and is not applied to this uniform example.
2026-10-10 · Independent AI editorial review: Independent root AI editorial review of all29 authored Measurement claims/conditions, displayed equations and37 origin relations. The original26 claims were read against41 primary source sections; all5 requested corrections and the3 affected source bindings were subsequently rechecked. Three actual scalar-nonlinear/vector/dynamic applications were read, independently differentiated or integrated, and checked against the specific additional source material described below. This is reference-content review, not human specialist certification, instrument certification or experimental replication. Full-field sign-off waits for the combined final conceptual matrix. Receipt: measurement-independent-content-review.1.
Entry details
Permanent link to this entry ↗- Record identity
- measurement-nonlinear-square-example
- Revision
- university-core.1
- Structured reference record
- University core expansion