Let $X$ be a standard normal variate. Consider another variate $Y$ such that
$$Y = \begin{cases} -X & \text{for $-2 < X< 2$} \\ X & \text{otherwise}. \end{cases}$$
I need to check whether $(X,Y)$ follow bivariate normal and correlation between $X$ and $Y$ is $1$.
My attempt: Since negative of a normal random variable is also normal, $Y$ is identical to $X$ and therefore their correlation is $1$. Since marginal are normal variates, $(X,Y)$ follow bivariate normal.
Am I right?
$X$ and $Y$ are not bivariate normal. If they were, then $X+Y$ would have be normal (or degenerate). But $X+Y=0$ on a set of positive probability (namley $\{|X| <2\}$) and it is not the zero random variable. Covariance can be computed explicitly using normal density. $X=EY=0$ so you only have to compute $EXY=-EX^{2}I_{\{|X| <2\}} +EX^{2}I_{\{|X| >2\}}$. I leave this to you.