Prove that

\[T^{-1} \frac{2}{9} \int_{-\infty}^{\infty} t^{2} \exp \left(-\frac{1}{4} t^{2}ight) d t=\frac{2}{3} \pi^{1 / 2} ho n^{-1 / 2},\]

and

\[T^{-1} \frac{1}{18} \int_{-\infty}^{\infty}|t|^{3} \exp \left(-\frac{1}{4} t^{2}ight) d t=\frac{2}{3} ho n^{-1 / 2},\]

where \(T=\frac{4}{3} ho^{-1} n^{1 / 2}\).