Probabilities of Bivariate Normal Distribution

Question

I have the following normal distribution that all of the parameter are known $$\begin{pmatrix} X_1\\ X_2\\ \end{pmatrix} \sim N\left[\begin{pmatrix} \mu_1 \\ \mu_2 \\ \end{pmatrix},\begin{pmatrix}\sigma^2 & \rho \\ \rho & \sigma^2\\\end{pmatrix} \right]$$

$1.$ $\Bbb P(X_1\le\mu_2) = \phi(\frac{\mu_2 - \mu_1}{\sigma})$ is that correct?

$2.$ How do I calculate $\Bbb P(X_1< X_2)$

Answer 1

1. Correct!

2. It is easy to verify that the marginal distributions are

$X_1\sim N(\mu_1;\sigma^2)$

$X_2\sim N(\mu_2;\sigma^2)$

with $Cov(X_1;X_2)=\rho$

Then it is immediate to calculate

$\mathbb{P}[X_1<X_2]=\mathbb{P}[X_1-X_2<0]$

via distribution of $Z=X_1-X_2$ that is known