# Distribution of X+Y of a bivariate normally distributed (X,Y)

#### endlessend25

##### New Member
I need to give the distribution of $X+Y$ and $X-Y$ knowing that $(X,Y)$ is bivariate normally distributed with marginal means 1, marginal variances 1 and correlation $p=2$.Is it right that the marginal mean are simply $E(X)=1$ and $E(Y)=1$ ?I don't see where I should start knowing that.

#### BGM

##### TS Contributor
First of all, when you are talking about random vector, its mean is also a vector of the same dimension. In the context it seems to indicate that both $$X$$ and $$Y$$ has a marginal mean of $$1$$, or equivalently the mean vector $$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$$ and so you should be correct. The marginal means are just the corresponding components in the mean vector.

One more thing is that the correlation is a bounded quantity in $$[-1, 1]$$. So the correlation cannot be $$2$$. Moreover, since the marginal variances are $$1$$ too, so the absolute value of covariance $$|Cov[X, Y]| = |\rho|\sigma_X\sigma_Y \leq 1$$, i.e. also bounded by $$1$$