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X and Y have continous distribution, the joint distribution is

$f_{X,Y}(x,y)=\frac{1}{2\pi\sqrt[]{1-p^{2}}}e^{-\frac{1}{2(1-p^2)}(x^2+y^2-2pxy)}$, ($p$ is a constante)

We need to find the marginal densiety $f_X$ of X.

I know we have to integrate over y:

$\int f_{X,Y}(x,y)dy=\int \frac{1}{2\pi\sqrt[]{1-p^{2}}}e^{-\frac{1}{2(1-p^2)}(x^2+y^2-2pxy)}dy$

but i can't seem to solve this can anyone help?

Anabella Woud
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    You should recognize this as a bivariate normal distribution with a certain mean and covariance. Then, you can read off the marginals (which are Gaussian with certain means and variances). – Batman Jun 12 '15 at 14:45
  • If we complete the square and forget about the constant in front, the problem comes down to finding $\int_{-\infty}^\infty e^{-((y-px)^2+(1-p^2)x^2))/(2(1-p^2))},dy$. Let $y-px=u\sqrt{1-p^2}$. – André Nicolas Jun 12 '15 at 14:57

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