# Probability of intersection of two absolute correlated normal random variables

Let $$(X_1,X_2)\sim \mathcal{N}_{2}(\boldsymbol{\mu},\mathrm{\Sigma})$$, where $$\boldsymbol{\mu} = (\mu_1,\mu_2)^{T}$$ and $$\mathrm{\Sigma}$$ is an appropriate variance matrix. Then how to find $$\mathbb{P}(|X_1|>c\cap |X_2|>c)$$?

• can't you just integrate the Gaussian over $x_1$ and $x_2$, both over the intervals $(-\infty,-c)$ and $(c,\infty)$? Commented Jul 25, 2021 at 12:45
• @CarloBeenakker I think no. Commented Jul 25, 2021 at 12:57
• you mean "no" like "this integral has no closed form" --- but that should not stop you from evaluating it numerically... Commented Jul 25, 2021 at 19:25
• Is it correct to write $\mathbb{P}(|X_1|>c\cap |X_2|>c) = 1 - \mathbb{P}(|X_1|\leq c)-\mathbb{P}(|X_1|\leq c) + \mathbb{P}(|X_1|\leq c\cap |X_2|\leq c)$? Commented Mar 23, 2022 at 14:28

Apparently, this probability cannot be expressed in closed form even when the means of the $$X_i$$'s are $$0$$ and their variances are $$1$$. At least, Mathematica cannot do anything with this probability:
• you would need a closed form for the indefinite integral of $e^{-x^2}\,\text{erf}(a+x)$, which is not available for $a\neq 0$. Commented Jul 25, 2021 at 19:25