Expected dot product of two random vectors.

Question

Consider random vectors $X,Y\in \mathbb{R}^d$ on a unit-euclidean-ball $\big($i.e, $||X||_2=||Y||_2=1\big)$, and a matrix $M \in \mathbb{R}^{d\times d}$. How to calculate the following expectation: $$\mathbb{E}\big[\big|X^TMY\big|\big]$$

Edit: Added the modulus sign

Thanks in advance!

Answer 1

Assume $X,Y$ are independent random vectors, chosen uniformly from the standard unit $d$-sphere.

Fix $Y$, and let $Z=MY$.

Then $Z$ is just some fixed vector in $\mathbb{R}^d$.

By symmetry, for $X$ chosen uniformly at random from the standard unit $d$-sphere, $$E[X\cdot Z] = E[(-X)\cdot Z] = E[-(X\cdot Z)]= -E[X\cdot Z]$$ hence $E[X\cdot Z]=0$.

Therefore $E[X^TMY]=0$.