What is the distribution of X+Y when (X,Y) has bivariate normal distribution.

Question

What is the distribution of X+Y when (X,Y) has bivariate normal distribution. I want to know the distribution of X+Y when (X,Y) has bivariate normal distribution and how to derive it.

Answer 1

Multivariate normal distribution has the following property:

If $X:p\times 1\sim N_{p}(\mu,\Sigma)$, then for any vector $\alpha:p\times 1$ of constants, $\alpha'X\sim N_{1}(\alpha'\mu, \alpha'\Sigma\alpha)$.

So, the distribution is univariate normal, with $$E(X+Y)=\alpha'\mu$$ and $$Var(X+Y)=\alpha'\Sigma\alpha$$ where $\alpha'=(1,1)$.

Answer 2

Let $X$ and $Y$ be jointly distributed as

$\left(\begin{array}{} X \\ Y \end{array} \right)\sim \mathcal N\left[\left(\begin{array}{} \mu_x \\ \mu_y \end{array} \right), \left(\begin{array}{} \sigma_x^2 & \rho \sigma_x\sigma_y\\ \rho \sigma_x\sigma_y & \sigma_y^2 \end{array} \right)\right]$

$\rho$ is the correlation coefficient. If $X$ and $Y$ are independent, then $\rho=0$.

The expected value of $Z=X+Y$ is $\mu_x+\mu_y$.

The variance of $Z$ is $Var(Z)=Var(X)+Var(Y)+2\cdot \rho \sigma_x\sigma_y$

Therefore $Z\sim\mathcal N\left( \mu_x+\mu_y,Var(X)+Var(Y)+2\cdot \rho \sigma_x\sigma_y\right)$

Thus the pdf is

$f_Z(z)=\frac{1}{\sqrt{2\pi}}\cdot \large{\frac{1}{\sqrt{\sigma_x^2+\sigma_x^2+2\cdot \rho \sigma_x\sigma_y}}}\cdot e^{\Large{-\frac{1}{2}\left(\frac{z-\mu_x-\mu_y}{\sqrt{\sigma_x^2+\sigma_x^2+2\cdot \rho \sigma_x\sigma_y}} \right)^2}}$