Computing the expectation of the outer product of two Gaussian vectors

Question

Let $a$ have iid $\mathcal{N}(0,I_n/n)$ entries and fix $x,y \in \mathbb{R}^n.$ I would like to compute $$\mathbb{E} \left[1_{\{a \cdot x > 0\}}1_{\{a \cdot y > 0\}}aa^T\right]$$ where $1_A$ is the indicator function on the event $A$. I know that $\mathbb{E}\left[aa^T\right] = I_{n \times n}/n$ since the off diagonal elements ($i \neq j$) are $\mathbb{E}[a_ia_j] = \mathbb{E}[a_i]\mathbb{E}[a_j] = 0$ by independence and the diagonal elements are \begin{align*} \mathbb{E}[a_i^2] = \mathbb{E}\left[\left(a_i - \mathbb{E}[a_i]^2\right)^2\right] = Var[a_i] = 1/n. \end{align*} However, I am uncertain as how to proceed with the indicator functions involved. I first tried to consider the case when $x = e_1$ and $y = \cos\theta e_1 + \sin \theta e_2$ where $\theta$ is the angle between $x$ and $y$. Then the indicators become $1_{\{a_1 > 0\}}$ and $1_{\{a_1\cos\theta + a_2\sin\theta > 0\}}$ so $\mathbb{E}[1_{\{a_1 > 0\}}] = \mathbb{P}(a_1 > 0) = 1/2$ but I don't know how to compute the probability of the second event. Can anyone provide any hints in the right direction?

Answer 1

This is really a 2 dimensional problem. Rotate coordinates such that $x_i=y_i=0$ for all $i\ge 3$. The expectation will be the sum of the interesting part depending on $(a_1,a_2)$ and a multiple of the identity matrix coming from the outer product of $(a_3,\ldots,a_n)$. For the interesting part, write $(a_1,a_2) = R (\cos \theta, \sin \theta)$, where $R$ and $\theta$ are independent, with known distributions. The restrictions $a\cdot x > 0$ and $b\cdot y>0$ translate into a range restriction on $\theta$. So the upper 2 by 2 corner of the answer is of form $$\frac{ER^2}{2\pi} \int_{\theta_0}^{\theta_1} \left( \begin{matrix}\cos^2\theta & \cos\theta\,\sin\theta\\\cos\theta\,\sin\theta&\sin^2\theta\end{matrix}\right)\,d\theta.$$