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

Chapter 3, Problem 22

(choose chapter or problem)

If  is an unbiased measurement of a true value \(\mu_{x} \text { and } U(X)\) is a nonlinear function of , then in most cases  is a biased estimate of the true value \(U\left(\mu_{x}\right)\). In most cases this bias is ignored. If it is important to reduce this bias, however, a bias-corrected estimate is \(U(X)-(1 / 2)\left(d^{2} U / d X^{2}\right) d_{x}^{2}\). 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 \(r=3.0 \pm 0.1 \mathrm{~cm}\)


a. Estimate the area
, and find the uncertainty in the estimate, without bias correction.
b. Compute the bias-corrected estimate of
.
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.

Equation Transcription:

Text Transcription:

\mu_x and U(X)

U(\mu_x)

U(X)-(1/2)(d2U / dX2)dx2

U(X)

r=3.0 \pm 0.1 cm

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