I am really stuck on this problem from the Concentration Inequalities Book, from Lugosi, Massart, Boucheron:
Let Z be $\chi$-distributed, with d degrees of freedom (ié, $Z^2 \sim\chi^2(d)$). Then I want to show
$$ \sqrt{d}-1 \leq \mathbb E[Z]\leq \sqrt{d} $$
The upper bound is fairly easy, since $\mathbb E[Z] \leq \sqrt{\mathbb E[Z^2]}$. For the lower bound, however, i can't seem to work it out.
For some context, the chapter's theme is the Efron-Stein inequality. My idea is to use the inequality to get some upper bound for the $\mathbb Var[Z]$ and use this upper bound to get the lower bound.
Say, if I managed to prove $\mathbb Var[Z]\leq A$ then I could make:
$$\mathbb Var[Z] = \mathbb E[Z^2] - \left(\mathbb E[Z]\right)^2\leq A$$
from here, I would get $\mathbb E[Z] \geq \sqrt{k - A}$, which seems to be close enough.
Anyway, does anyone have any idea on how to use the Efron-Stein Inequality to get that Lower Bound?