$(20\%)$ Let $x$ denote a random variable. Define

$G_x(t)=E(t^x)$

for tinS, where $S$ is a set which ensures the existence of $(1).$ Now, we specify the pdf $x$ as

follows:

$f(x)=\frac{1}{\sqrt[\phantom{2}]{2}}$exp$\{-\frac{(x-{)}^2}{2{}^2}\},$xinR

$($a$)$ Derive $G_x(t)$ under the $pdf(2).$

$($b$)$ Use $($a$)$ to derive $E(x)$ and var$(x).$