If Y = f (x,x2,..., Xx), where X, is estimated by x; with standard uncertainty u(x) and the X; are statistically independent, what is the estimate y of Y and its standard uncertainty u(y) as given by the law of propagation of uncertainty (univariate LPU)? if Y = f(x), where X = (X1, X3,..., XN)' is estimated by x with covariance matrix V x, what is the estimate y of Y = (Y..Y.... Ym) and its covariance matrix V, as given by the law of propagation of uncertainty (multivariate LPU)? = р The mass of a spherical artefact is given by M = npr where r is the radius and p its density. Express the uncertainties associated with M, - M, where M, and M, are the masses of spherical artefacts made from the same material in terms u(re), u(r.), and u(p). Finally, suppose that a third spherical artefact with radius r known very accurately but with unknown density P3 is compared in a mass balance with Ma and that an estimate y of M, - M, and associated uncertainty u(y) is produced. Derive an estimate of Pa, its associated uncertainty u(p), and the 2 x 2 variance matrix V associated with p, and pz. Answers can be expressed in terms of sensitivity coefficients and uncertainties.
If Y = f (x,x2,..., Xx), where X, is estimated by x; with standard uncertainty u(x) and the X; are statistically independent, what is the estimate y of Y and its standard uncertainty u(y) as given by the law of propagation of uncertainty (univariate LPU)? if Y = f(x), where X = (X1, X3,..., XN)' is estimated by x with covariance matrix V x, what is the estimate y of Y = (Y..Y.... Ym) and its covariance matrix V, as given by the law of propagation of uncertainty (multivariate LPU)? = р The mass of a spherical artefact is given by M = npr where r is the radius and p its density. Express the uncertainties associated with M, - M, where M, and M, are the masses of spherical artefacts made from the same material in terms u(re), u(r.), and u(p). Finally, suppose that a third spherical artefact with radius r known very accurately but with unknown density P3 is compared in a mass balance with Ma and that an estimate y of M, - M, and associated uncertainty u(y) is produced. Derive an estimate of Pa, its associated uncertainty u(p), and the 2 x 2 variance matrix V associated with p, and pz. Answers can be expressed in terms of sensitivity coefficients and uncertainties.
