Suppose we want to find a circle to fit the data points. The error of the data point can be calculated as
$e_k=||\vec{x}_k-\vec{x}_c||^2-r^2$ ------------------(1)
where the $\vec{x}_k$ is the coordinate of the data point, the $\vec{x}_c$ and $r$ is the coordinate of the center and radius of the fitted circle.
Assuming that the data points can be modeled as
$\vec{x}_k=\hat{\vec{x}}_c+\vec{m}$------------------------------(2)
where $\hat{\vec{x}}_k$ is the coordinate of the point corresponds to a data point (on the fitted circle), and $\vec{m}$ is independent, isotropic Gaussian noise: $\vec{m}$~$\mathscr{N}(0, \sigma^2\textbf{I})$. Probability density function:
$p(\vec{m})=\frac{1}{2\pi\sigma^2}e^{-\frac{\vec{m}^T\vec{m}}{2\sigma^2}}$----------------------(3)
The question is: what is the distribution of the error $e_k$?
By substituting $\vec{x}_k$ in (1) with (2), I got
$e_k=||\vec{m}||^2+2r||\vec{m}||$-------------------(4)
However, since the two terms of $e_k$ are not independent, we cannot simply add the distribution together. So how can I determine the distribution of $e_k$?