There are two random variables $X$ ~ $N(0,1)$, and $Y$ ~ $N(0,1)$ with correlation $0.5$.
How to find $P(0 < Y<X)$?
I stuck on this interview question. What confuses me is the correlation $0.5$ when $X$ and $Y$ are two standard normal variables. Any hint, please?
Correlated normal variables can be interpreted geometrically. If we sampled independent normal variables $X_1, X_2 \sim N(0,1)$, then the joint distribution of $(X_1, X_2)$ is rotationally symmetric, and for any unit vector $\textbf u$, measuring $(X_1,X_2)$ in the direction of $\textbf u$ produces another standard normal: $$(X_1, X_2) \cdot \textbf u \sim N(0,1).$$ Moreover, if we measure in perpendicular directions, we get independent random variables (in particular, measuring in the direction of $\textbf e_1 = (1,0)$ and $\textbf e_2 = (0,1)$ recovers $X_1$ and $X_2$). But we can also measure in directions that aren't perpendicular. If we take two directions separated by an angle of $\theta$: \begin{align} X = (X_1,X_2) \cdot (1,0) &= X_1 \\ Y = (X_1,X_2) \cdot (\cos \theta, \sin \theta) &= X_1 \cos \theta + X_2 \sin \theta \end{align} we get $\mathbb E[XY] = \mathbb E[X_1^2] \cos \theta + \mathbb E[X_1 X_2] \sin\theta = \cos \theta$. In particular, if $\theta = \frac\pi 3$, the correlation between $X$ and $Y$ is $\cos \theta = \frac12$.
So now we know $X$ and $Y$ can be obtained by measuring $(X_1,X_2)$ in two directions that are an angle of $\frac\pi3$ apart: say, $0$ and $\frac\pi3$ radians from the $x$-axis. What does that buy us? Well, both measurements are positive if the vector $(X_1, X_2)$ points in a direction ranging from $\frac\pi3 - \frac\pi2 = -\frac\pi6$ to $0 + \frac\pi2 = \frac\pi2$, which measures $\frac{2\pi}{3}$ in total. By symmetry, this happens $\frac13$ of the time. Again by symmetry, $X>Y$ half the time, so overall we get $\Pr[X>Y>0] = \frac16$.
This is pictured above. The red and blue vector are the direction in which we measure $X$ and $Y$, respectively. The six regions around the circle where $(X_1,X_2)$ can land correspond to six possible orderings of $0$, $X$, and $Y$.