\end{cases}In particular, we see that it is severely skewed to the right: it is much more likely for \lambda_{max} to be much larger than 2\sigma\sqrt{N} than to be much smaller. This could be intuited by seeing that the limit distribution is the semicircle law, so there is "repulsion" from the bulk of the distribution, forcing \lambda_{max} to be not much smaller than 2\sigma\sqrt{N}.
