Integrating a function over a bivariate normal distribution












1












$begingroup$


Given a bi-variate Normal distribution
$$N(x_1, x_2) = frac{1}{2pisigma_1sigma_2sqrt {1-rho^2}}e^{-z/2(1-rho^2)}$$



$$ z = frac{(x_1-mu_1)^2}{sigma_1^2}- frac{2rho(x_1-mu_1)(x_2-mu_2)}{sigma_1sigma_2} + frac{(x_2-mu_2)^2}{sigma_2^2} $$



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



I want to calculate the following integral:



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



How should I solve it?



Thanks in advance!






calculus integration statistics







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$endgroup$

















    1












    $begingroup$


    Given a bi-variate Normal distribution
    $$N(x_1, x_2) = frac{1}{2pisigma_1sigma_2sqrt {1-rho^2}}e^{-z/2(1-rho^2)}$$



    $$ z = frac{(x_1-mu_1)^2}{sigma_1^2}- frac{2rho(x_1-mu_1)(x_2-mu_2)}{sigma_1sigma_2} + frac{(x_2-mu_2)^2}{sigma_2^2} $$



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



    I want to calculate the following integral:



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



    How should I solve it?



    Thanks in advance!






    calculus integration statistics







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Given a bi-variate Normal distribution
      $$N(x_1, x_2) = frac{1}{2pisigma_1sigma_2sqrt {1-rho^2}}e^{-z/2(1-rho^2)}$$



      $$ z = frac{(x_1-mu_1)^2}{sigma_1^2}- frac{2rho(x_1-mu_1)(x_2-mu_2)}{sigma_1sigma_2} + frac{(x_2-mu_2)^2}{sigma_2^2} $$



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



      I want to calculate the following integral:



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



      How should I solve it?



      Thanks in advance!






      calculus integration statistics







      share|cite|improve this question











      $endgroup$




      Given a bi-variate Normal distribution
      $$N(x_1, x_2) = frac{1}{2pisigma_1sigma_2sqrt {1-rho^2}}e^{-z/2(1-rho^2)}$$



      $$ z = frac{(x_1-mu_1)^2}{sigma_1^2}- frac{2rho(x_1-mu_1)(x_2-mu_2)}{sigma_1sigma_2} + frac{(x_2-mu_2)^2}{sigma_2^2} $$



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



      I want to calculate the following integral:



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



      How should I solve it?



      Thanks in advance!






      calculus integration statistics




      calculus integration statistics






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Sep 30 '16 at 17:47









      Michael Hardy

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      asked Sep 30 '16 at 17:44









      user373921user373921

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