Before answering this question, construct the function, rpareto, in $R$ using the following code:

rpareto $=$ function $,$ alpha, $l$ ambda$)1$

$rp=1$ ambda $***((1-$runif$(n){)}^{{?}^{{?}^{?}}}(-\frac{1}{})-1)$

rp$)$

The rpareto function generates a random sample of size $n$ from a two$-$parameter Pareto distribution with parameters $$ and $.$

$($i$)$ Generate, using rpareto, a random sample of size $25,000$ from the two$-$parameter Pareto distribution with parameters $=3$ and $=1,$ assigning the simulated values to a vector called A$\_$vec. You should set a random number generator seed of NNN

$1$

before generating A$-$vec, where NNN is the last three digits of your student ID$.$ Use the $R$ function, head$(),$ to display the first five values of A$\_$vec in your answer script. $[20]$

$($ii$)$ Construct an $R$ function, called A$\_$exceed$\_$u$,$ with two arguments, A and $u,$ that returns only the non$-$zero entries of a vector, $E,$ where vector $E$ is of length $25,000,$ with entries defined as:

$E_i=$max$\{A_i-u,0\}$

Use the $R$ function, head$(),$ to display in your answer script the first five values of A$\_$exceed$\_$u when $A=A_{v}ec$ and $u=1.$

$($iii$)$ Construct another R function, called $F_{-}u,$ with one argument, A$\_$greater$-$than$\_$u$,$ that returns a vector of length $101,$ containing the probabilities:

$P(A_i-ux|A_i>u),$ where $x0,$

for values of $x$ from $0$ to $10$ inclusive, at intervals of $0\mathrm{.}1.$

Use the $R$ function, head$(),$ to display in your answer script the first five values of $F_u$ when $A_{g}reate{r}_{t}ha{n}_u=A_{e}xcee{d}_u$ with arguments $A=A_{v}ec$ and $u=1.[20]$

$($iv$)$ Plot, on a single graph, four line graphs of the values of $F_{-}u$ against $x$ for the values of $x$ specified in part $($iii$)$ when A$\_$greater$\_$than$\_$u $u$ A$\_$exceed$\_$u$,A=A_{v}ec$ and $u=1,2,$ Y and $4.$ You should use separate colours to identify each line graph and you should clearly specify which value of $u$ is represented by each of the four line graphs.

$[20]$

$($v$)$ Comment on the graph produced in part $($iv$).$

Hint: You are given that if $xPareto(,),$ then the threshold exceedance $x-u|x||>u$ is distributed as Pareto $(,+u).$

$[20]$