Suppose $X \in [-1, 1]$ is a random variable, and $A$ is a positive semi-definite matrix. Is $E[X]E[AX] - E[XAX]$ positive semi-definite?

Question

Let $X \in [-1, 1]$ denote a random variable. Let $A \in \mathbb{R}^{n \times n}$ denote a random positive semi-definite (PSD) matrix. $X$ and $A$ are dependent. Let $E$ denote expectation.

My question is:

Does $$v^TE[X]E[AX]v \geq 0 \Longrightarrow v^TE[XAX]v \geq 0?$$

By Hölder's inequality, for two random variables $X, Y$, $$E[XY] \leq E(|X|)E(|Y|)$$ Therefore, I'm wondering if something similar would hold in case where $Y$ is a matrix.

Answer 1

Well, since $A$ is positive semidefinite, then $$ v^{\top}\mathsf{E}[X^2A]v=\mathsf{E}[v^{\top}X^2Av]\ge 0 $$ because $v^{\top}X^2Av\ge 0$ for any $v\in \mathbb{R}^n$.