Given a bi-variate Normal distribution $$N(x_1, x_2) = \frac{1}{2\pi\sigma_1\sigma_2\sqrt {1-\rho^2}}e^{-z/2(1-\rho^2)}$$

$$ z = \frac{(x_1-\mu_1)^2}{\sigma_1^2}- \frac{2\rho(x_1-\mu_1)(x_2-\mu_2)}{\sigma_1\sigma_2} + \frac{(x_2-\mu_2)^2}{\sigma_2^2} $$

$$\rho = \operatorname{corr}(x_1, x_2) $$

I want to calculate the following integral:

$$ \int_0^\infty\int_{-\infty}^0 x_1x_2N(x_1,x_2) \, dx_1 \, dx_2 $$

How should I solve it?

Thanks in advance!


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