I'm working on the following problem:
Let $G = SO(n)$ for the group of rotation matrices on $\mathbb{R^n}$ with $n \geq 2$. Consider the Haar measure $\mu $ on $G$, and let $e_i$ denote the standard unit vector in $\mathbb{R}^n$.
Evaluate the following integrals: $$ E_{ij} := \frac{1}{\mu(G)}\int_G(e_i \cdot Ue_j)d\mu (U)$$ $$ V_{ij} := \frac{1}{\mu(G)}\int_G|(e_i \cdot Ue_j) - E_{ij}|^2d\mu (U)$$ $E_{ij}$ and $V_{ij}$ give the expectation and variance, respectively, of the $(i,j)$th entry of a randomly chosen rotation matrix $U$.
My attempt:
I want to use this result about orbital averages:
$$ \frac{1}{\mu(G)}\int_{SO(n)}f(U_x)d\mu (U) = \frac{1}{\sigma(S^{n-1})}\int_{S^{n-1}}f(y)d\sigma (y)$$
where $x \in S^{n-1}$ (i.e. an arbitrary unit vector) and $f: S^{n-1} \to \mathbb{R}$.
In the case of $E_{ij}$, we have $$ E_{ij} = \frac{1}{\sigma(S^{n-1})}\int_{S^{n-1}}(e_i \cdot y) d\sigma (y) = \frac{1}{\sigma(S^{n-1})}\int_{S^{n-1}}y_i d\sigma (y) =0$$ For $V_{ij}$, we have $$\frac{1}{\mu(G)}\int_G|(e_i \cdot Ue_j) - E_{ij}|^2d\mu (U) =\frac{1}{\sigma(S^{n-1})} \int_{S^{n-1}} |e_i \cdot y|^2d\sigma (y) \\ =\frac{1}{\sigma(S^{n-1})} \int_{S^{n-1}} |y_i|^2d\sigma (y)$$ How do I proceed from here in evaluating this interval over the unit sphere? Any advice or hints would be appreciated!