Pattern Recognition

In order for the estimator to be unbiased, the expected value must be equal to the true value, which gives

$E\left\{\widehat{{\sigma }^2}\right\}={\sigma }^2\iff \frac{N}{\alpha -N}=1\iff \alpha =2N.$

(4.6)The expected value of the estimator ̂μ is

$E\left\{\widehat{{\sigma }^2}\right\}={\sigma }^2\iff \frac{N}{\alpha -N}=1\iff \alpha =2N.$

It is unbiased if

$E\left\{\widehat{{\sigma }^2}\right\}={\sigma }^2\iff \frac{N}{\alpha -N}=1\iff \alpha =2N.$

A.5Chapter 5

(5.1)The inverse mapping $x=A^{-1}\left(y\right)=\pm \sqrt{y}$y to x is ill-posed.

(5.2)The system of equations to solve this problem is over-determined, since there are more data points (N < 2) than degrees of freedom (a,b). This means that in general there is no solution that interpolates all data points. Therefore, the problem is ill-posed.

The following variation is well-posed.

Find the parameters a, b∈ of a straight line y= a x+ b