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I have a vector $\mathbf{v} = [ v_x , v_y ] ^T$ whose PDF is Gaussian, thus it has an associated covariance matrix to represent its uncertainty (which is zero mean):

$P = \begin{bmatrix} \sigma_x^2 & \sigma_x \sigma_y \\ \sigma_x \sigma_y & \sigma_y^2 \end{bmatrix}$

How can I calculate the uncertainty of the 2-norm $\sigma_v$ for $v = \sqrt{ v_x^2 + v_y^2}$ ?

The closest I can find is the formula for uncertainty when $\sigma_x \sigma_y = 0$:

$\sigma_v = \sqrt{ \left ( \frac{\partial v}{\partial v_x} \sigma_x \right )^2 + \left ( \frac{\partial v}{\partial v_y} \sigma_y \right )^2 }$

.. However this does not include the covariance term.

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