Let's say I have a normally distributed random variable $X_1$ with known standard deviation $\sigma_1$ and $E[X_1]$ is $0$. Let's say I have another variable with known standard deviation $\sigma_2$ and $E[X_2]$ is $-V$. Let's say the are correlated with $\rho = .9$.
This implies the $\beta$ (defined by $X_2 = \alpha + \beta X_1$) between $X_1$ and $X_2$ is $\beta =\rho \Sigma_2/\Sigma_1$.
I want to know the probability that the value $[X_1]$ is less than or equal to $[X_2]$, i.e., $P(X_2 \ge X_1)$.
I tried to conceptualize this as concentric circles on an $X_1-X_2$ plane... but I'm not sure how to combine these variables with a known correlation.