I'm studying dimensionality reduction (SVD in particular), and I saw the following question:
Assume we have a vector $x \in \mathbb R^d$, and consider $F(x)=s^t x$ , where $s$ is a $d$-dimensional random vector with entries drawn uniformly independently from $[-1,1]$.
What is the value of $\mathbb E[F(x)^2]$?
I'm starting now from zero, so I need to study more. The exercise asks for a formal proof, but I wish to just understand the philosophy.
I see many questions in which valued are drawn uniformly and independently from $[-1,1]$. Is this distribution used due to its symmetrical range? Is the expectation in this range often $1/2$?