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

#1
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
#2
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 \( 1 \)