Let $M \in \mathbb{R}^{n*n}$ be a unitary matrix such that it's values are bounded by C. Let $x\in \mathbb{R}^{n}$ be an s-sparse vector. Let $R\in \mathbb{R}^{k*n}$ be a matrix with rows randomly chosen $k$ from $M$ with repetition, multiplied by $\sqrt{n/k}$. Compute the expectation and variance of $||Rx||^2 $ as a function of $||x||^2,k,s,C$.
So far I've been able to calculate the expectation, as M is unitary it is norm preserving, so the expectation of each entry in $Rx$ is $\sqrt{n/k}*x_i$, which after opening the norm and using linearity of expectation, implies. $$E||Rx||^2=||x||^2$$ As for variance, I'm trying to calculate $E||Rx||^4$ and derive the variance from there. I'm having trouble understanding how to use $s,C$ in the calculation.