$($a$)$ A random variable $x$ has pdf

$f(x)=\{\begin{array}{ccc}{e}^{-x}& if& x>0\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

where $>0.$

i$.$ Find the probabilities $P(|x-0\mathrm{.}5|>2)$ and $P(xinQ)$ where $Q$ is the set of

rational numbers.

ii$.$ Find the cumulative distribution function of $x.$

iii. Using c$.$d$.$f$.$ find $E(x)($Check whether all the necessary assumptions hold true$).$