#1
the radius of a circle is measured with an error of measurement which is distributed normal with mean \(0\) and variance \(\sigma^2\),\(\sigma^2\) unknown.Given \(n\) independent measurements of the radius , find an unbiased estimator of the area of the circle.

By using *Maximum Likelihood Estimator* I found

\(\hat\sigma^2=\frac{\sum r_i^2}{n}\)

where \(r\) is the radius of the circle and \(r\sim N(0,\sigma^2)\).

Then i am stucked to find the unbiased estimator of the area of the circle
 

Dragan

Super Moderator
#2
the radius of a circle is measured with an error of measurement which is distributed normal with mean \(0\) and variance \(\sigma^2\),\(\sigma^2\) unknown.Given \(n\) independent measurements of the radius , find an unbiased estimator of the area of the circle.

By using *Maximum Likelihood Estimator* I found

\(\hat\sigma^2=\frac{\sum r_i^2}{n}\)

where \(r\) is the radius of the circle and \(r\sim N(0,\sigma^2)\).

Then i am stucked to find the unbiased estimator of the area of the circle

What is the expected value of: \(\pi\frac{\sum r_i^2}{n}\),
while noting that the sample variances of the n radius measures will have expected value of sigma^2.