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For a 2D Normal distribution $N(0, \left[ \begin{array}{cc} 1 & -1/4 \\ -1/4 & 1 \end{array} \right])$, I am now trying to maximize its density function over $\{ x\geq 10, y \geq 10 \}$.

My intuition is that the optimal solution is $(10,10)$. My problem is that I am not sure which way to find the optimal solution.

Thanks!

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up vote 4 down vote accepted

The density of your distribution is $$ \exp \left( {-1 \over 2(1-1/16)} \left[ x^2 + y^2 + {xy \over 2} \right] \right)$$ multiplied by a normalizing constant. (See Wikipedia; you have $\mu_x = \mu_y = 0, \sigma_x = \sigma_y = 1, \rho = -1/4$.)

So your problem reduces to minimizing the quadratic $x^2 + y^2 + xy/2$ over $\{ x \ge 10, y \ge 10 \}$. This function is increasing in both $x$ and $y$ and so is minimized at $x = y = 10$.

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