$(30$ points$)$ Suppose $x$ follows a triangle distribution with parameter $(-1,0,1).$

$($i$)$ Derive the pdf of $x.$ Provide the ranges of $x$ to receive full credit.

$(10$ points$)$

$($ii$)$ Derive the cumulative distribution function of $x.$ Provide the ranges of $x$ to receive full

credit.

$(10$ points$)$

$($iii$)$ Write an inversion algorithm $($pseudo$-$code$)$ that generates $x.$

$(10$ points$)$

$(5$ points$)$ Suppose in one of your upcoming class assignment, your classmate wants to model the

daily arrival process of passengers to the airport terminal using a Poisson distribution. However,

you don't agree with him$/$her$/$they$.$ List all reasons to support your argument. $($Receive full

credits if at least $2$ reasons are valid$)$