Consider a random variable $a$ and constant vector $b$ drawn from $\mathbb{R}^n$. Let $\theta$ be the angle between the two vectors. I am interested in finding expectation of $\mathbb{E}[||a||^2\sin^2\theta]$. Is it correct to assume that the norm "$||a||$" and the sine of angle "$\sin\theta$" are independent of each other?
Intuitively, I think they should be independent because knowledge of one does not tell anything about the other.
Another intuition I am thinking of that might justify it is using probabilistic graphical models. We can consider a causal graph of a vector $a$, its norm $||a||$ and the angle $\theta$ w.r.t $b$, such that vector $a$ is a common cause that leads to the norm and the angle. So using the common cause analysis for the graphical model, the norm and angle must be at least conditionally independent.
Is my intuition in the right direction? Is there a more rigorous mathematical way of proving this?
In general, they are not independent. Consider $b=(1,0)$ and $P(a=(1,0))=P(a=(0,2))=1/2$. Then obviously The angle is $0$ iff $\Vert a\Vert=1$.