Goal: computation of the probability mass associated to the interval (inner or outer).
Goal: computation of the probability mass associated to the interval (inner or outer).
Note that the diagonal matrix corresponds to $\lambda_1=\lambda_2$ and $\theta=0$.
A generic bivariate covariance matrix may be written as $$ \Sigma= R S R^T $$ where $S$ and $R$ denote the scaling and rotation of a diagonal covariance matrix. The scaling matrix $$ S=\begin{bmatrix} \lambda_1 & 0 \ 0 & \lambda_2 \end{bmatrix}$$ stretches the x-axis and the y-axis by the factors $\lambda_1>0$ and $\lambda_1>0$ respectively.
The matrix $$ R=\begin{bmatrix} \cos(\theta) & -\sin(\theta) \ sin(\theta) &\cos(\theta) \end{bmatrix}$$ performs a rotation of angle $\theta$.
Suggested manipulations:
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