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Define $x\sim\mathcal{N}(0,\Sigma)$ be $n$-dimensional Gaussian vector and two half-spaces $Q_1:=\{x\in\Re^n:a^\top x\ge 0\}, Q_2:=\{x\in\Re^n: b^\top x\ge 0\}$, where $a,b$ are unit vectors. What is the probability that $x$ falls into both half-spaces $P_x(x\in Q_1\cap Q_2)$?

We can conclude $P(x\in Q_1)=P(x\in Q_2)=1/2$ by symmetrical considerations, or by using the special case of $\mu=0$ from here. But in case of the intersection, the answer must clearly depend on the vectors $a,b$, as it is the case for the two extreme cases where $a=b$ and $a=-b$ leading to $P_x(x\in Q_1\cap Q_2)=1/2$ and $P_x(x\in Q_1\cap Q_2)=0$ respectively. My main question is, is it possible to state the probability in terms of $a,b$ only, and regardless of the covariance structure $\Sigma$?

kvphxga
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1 Answers1

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The probability that $x\in Q_1$ and $x\in Q_2$ depends on the joint probability distribution of Gaussian random variables $\langle a, x\rangle$ and $\langle b, x\rangle$. Note that the joint distribution of two centered Gaussian variables is uniquely determined by their covariance matrix. In our case, this matrix is $$\begin{pmatrix}a^T \Sigma a & a^T\Sigma b \\ a^T \Sigma b & b^T\Sigma b\end{pmatrix}$$

Further, it's easy to see that the answer depends only on the directions of $a$ and $b$ and not on their lengths. That is, the desired probability is determined by the correlation of random variables $\langle a, x\rangle$ and $\langle b, x\rangle$, which is

$$\rho = \frac{a^T\Sigma b}{\sqrt{a^T \Sigma a} \sqrt{b^T \Sigma b}}.$$

To compute the probability that $x\in Q_1 \cap Q_2$ in terms of $\rho$, assume that $a$ and $b$ are unit vectors in ${\mathbb R}^2$, the angle between them is $\arccos \rho$, and $x\sim {\cal N}(0, I_2)$. Then we get that $$\Pr(x\in Q_1\cap Q_2) = \frac12 - \frac{\arccos \rho}{2\pi}.$$

(The answer clearly depends on $\Sigma$.)

Yury
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