Problem I $(20$ marks$)$

I$)(8$ marks$)$ Show that the following bound holds for any $0$ and $0$ :

$Q(+)Q()exp(--\frac{{}^2}{2})$

Hint: Start with $Q(+)={\displaystyle \underset{+}{\overset{\phantom{}}{}}}$exp$(-\frac{x^2}{2})dx$ and use the change of variable $y=$

$x-.$

$(6$ marks$)$ Use the result of I$)$ to show that for all $0,$

$Q()\frac{1}{2}$exp$(-\frac{{}^2}{2})$

$(6$ marks$)$ Use the result of I$)$ to show that for all $0,$

$\frac{Q()}{\text{e}\text{x}\text{p}(-\frac{{}^2}{2})}\frac{Q()}{\text{e}\text{x}\text{p}(-\frac{{}^2}{2})}$