$($c$)$ A function $g$:$RR$ is a density of a probability distribution if and only if $($i$)g(x)0$ for

all xinR, and $($ii$)$

${\displaystyle \underset{-}{\overset{}{}}}g(x)dx=1.$

The density of the normal distribution with mean $5$ and standard deviation $3$ is $f$:$RR$

given, for xinR, by

$f(x)=\frac{e^{-\frac{(x-5{)}^2}{18}}}{\sqrt[\phantom{2}]{18}}.$

Verify that $f$ is a density. When examining the relation in $($ii$),$ please use integration

methods from this course.