Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.


share|cite|improve this question
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 $x = y = 10$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.