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Let $X \sim N(0, \Sigma)$ be a bivariate normal distribution with $\Sigma \in \mathbb{R}^{2 \times 2}$ and know mean of $(0,0)^\intercal$. I know how to calculate Fisher information matrix of $X$, but I would like to know if the inverse of the Fisher information matrix exists in this case.

I need this to show that, for a certain theorem, the regularity conditions are met.

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