I'm using the notations of the Erdős-Yau book. On page 11: Let $H$ be an $N \times N$ real symmetric or complex Hermitian Wigner matrix, where the upper triangle entries of $H$ are independent, the entries are centered $\mathbb{E}h_{ij}=0$, and the variances are $\mathbb{E}|h_{ij}|^2=N^{-1}$. Let $\lambda_1\leq\dots\leq\lambda_N$ denote the eigenvalues of $H$. The Wigner semicircle law states that for any fixed $a\leq b$ real numbers, we have $$ \lim_{N\to\infty}\frac{\#\{i: \lambda_i \in [a,b]\}}{N} = \frac{1}{2\pi}\int_a^b \sqrt{(4-x^2)_+}, $$ where $(\alpha)_+ := \max\{0,\alpha\}$ denotes the positive part of a number $\alpha$.
This implies that as $N$ goes to infinity, all the eigenvalues are in the interval $[-2,2]$. There are two proofs in the above linked book for this statement. The first uses the moment method, and the second uses the Stieltjes transform method, but from these proofs it is not clear for me, how $4$ appears in the integrand above. I'm pretty sure that there is a probabilstic reasoning for this, and it would not be surprising if it is because of some basic properties of moments for example, but I just can't figure it out.
Why the radius of the semicircle equals $2$ in the Wigner semicircle law?