Good morning, I can't find how to prove this inequality :
$F_d(x) ≥ (⌊d/2⌋ + 1)*F_2(x/(⌊d/2⌋ + 1))$
where $F_d(x) = P(\lVert Z_s \rVert^2 \leq x)$ with $Z_s \sim\mathcal{N}(0, I_d)$
I have the intuition we should group together two by two the component of the vector but then I don't know how to show it. I guess we have to use the fact that $F_d(x)$ is the distribution function of the $\chi^2(d)$ distribution.