$(20\%)$ Let $x$ denote a continuous random variable with the support $(-,).$ Define a

hazard function

$(x)=\underset{0}{lim}\frac{P(xx=x)}{}$

$($a$)$ Show that

$(x)=\frac{f_x(x)}{1-F_x(x)}$

where $f_x(x)$ and $F_x(x)$ are the pdf and $CDF$ of $x,$ respectively.

$($b$)$ Use $($a$)$ to show that ${\displaystyle \underset{-}{\overset{x}{}}}(t)dt=-ln\{1-F_x(x)\}.$

$($c$)$ Here we particularly specify $(x)=\frac{}{}{x}^{-1}$ for $x>0,$ where $$ and $$ are parameters.

Use $($b$)$ to determine the $CDF{F}_x(x).$

$($d$)$ Continue on $($c$),$ explain how to use the uniform distribution to generate the random

variable $x.$