If \(X\) is an unbiased measurement of a true value \(\mu_{X}\), and U(X) is a nonlinear function of \(X\), then in most cases \(U\) 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 \(U(X)-(1 / 2)\left(d^{2} U / d X^{2}\right) \sigma_{X}^{2}\). 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 \(r=3.0\pm0.1\mathrm{\ cm}\).

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.

Equation Transcription:

Text Transcription:

X

mu_X

U

U(mu_X)

U(X)-(1/2)(d^2 U / dX^2) sigma_X ^2

r = 3.0 pm 0.1 cm

A

Solution :

Step 1 of 3:

Given X is an unbiased measurement of a true value .

Then a bias-corrected estimate is .

Where U(X) is a nonlinear function of X.

We assume that the radius of a circle is measured to be .

Our goal is :

a). We need to estimate the area and the uncertainty in the estimate bias correction.

b). We need to compute the bias-corrected estimate of A.

e). We need to find is bias correction important in this case? Explain.

a).

Now we have to estimate the area and the uncertainty in the estimate bias correction.

The area of the circle is

Where and r=3.

Then

Therefore the area of the circle is 28.26.

The uncertainty in the estimate bias correction is

Where A=

Here we are differentiating with respect to r.

and

Therefore the uncertainty in the estimate bias correction is 1.8840.