If X is an unbiased measurement of a true value X, and U(X) is a nonlinear function

If X is an unbiased measurement of a true value μX, and U(X) is a nonlinear function of X, then in most cases U is a biased estimate of the true value U(μX ). In most cases this bias is ignored. If it is important to reduce this bias, however, a bias-corrected estimate is U(X) − (1/2)(d²U/dX²)σ²_X. In general the bias corrected estimate is not unbiased, but has a smaller bias than U(X). Assume that the radius of a circle is measured to be r = 3.0 ± 0.1 cm.

a. Estimate the area A, and find the uncertainty in the estimate, without bias correction.

b. Compute the bias-corrected estimate of A.

c. Compare the difference between the bias-corrected and non-bias-corrected estimates to the uncertainty in the non-bias-corrected estimate. Is bias correction important in this case? Explain.