Given the Irwin-Hall distribution with pdf, [f_{Y}(y)=frac{1}{2(n-1) !} sum_{k=0}^{n}(-1)^{k}left(begin{array}{l}n kend{array} ight)(y-k)^{n-1} operatorname{sign}(y-k)] where the sign function (operatorname{sign}(x))

Given the Irwin-Hall distribution with pdf,

\[f_{Y}(y)=\frac{1}{2(n-1) !} \sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}n \\k\end{array}\right)(y-k)^{n-1} \operatorname{sign}(y-k)\]

where the sign function \(\operatorname{sign}(x)\) is defined on \((-\infty, 0) \cup(0, \infty)\) as\[\operatorname{sign}(x)=\frac{|x|}{x}= \begin{cases}1 & \text { if } x>0 \\ -1 & \text { if } x<0\end{cases}\]

verify that the Irwin-Hall distribution has mean \(n / 2\) and variance \(n / 12\).