Let $(X, Y)$ have a normal distribution with mean $(\mu_X, \mu_Y)$, variance $(\sigma_X^2, \sigma_Y^2)$ and correlation $\rho$. I want to know the corresponding marginal densities.
All I found so far was the well-known density expressions for $X\sim N(\mu_X, \sigma_X^2)$ and $Y\sim N(\mu_Y, \sigma_Y^2)$, but isn't that just for $X \perp Y$? Shouldn't $\rho$ appear in the expressions? How do I calculate the marginals of any $X$ and $Y$? I think using the definition will end up in an integral that cannot be solved analytically…
Best Answer
Following up on whuber's comment, $$\begin{align} \left.\left.\frac{1}{2(1-\rho^2)}\right(x^2+y^2-2\rho xy\right) &=\left.\left.\frac{1}{2(1-\rho^2)}\right(x^2-\rho^2x^2+(y^2-2\rho xy + \rho^2x^2)\right)\\ &= \frac{x^2}{2} + \frac{1}{2}\left(\frac{y-\rho x}{\sqrt{1-\rho^2}}\right)^2 \end{align}$$ and so $$\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy = \frac{e^{-x^2/2}}{\sqrt{2\pi}} \int_{-\infty}^\infty \frac{e^{-(y-\rho x)^2/2(1-\rho^2)}}{\sqrt{1-\rho^2}\sqrt{2\pi}}\,\mathrm dy.$$ I will leave it to the OP to complete the details and determine whether $\rho$ disappears or not when the integral is evaluated.