$>$using r programming x

$[1]2\mathrm{.}05360321\mathrm{.}69795512\mathrm{.}57784300\mathrm{.}39734970\mathrm{.}53014611\mathrm{.}25797701\mathrm{.}25344980\mathrm{.}85225233\mathrm{.}6920431$

$[10]0\mathrm{.}38179482\mathrm{.}24146161\mathrm{.}54953981\mathrm{.}18535571\mathrm{.}37316760\mathrm{.}97026812\mathrm{.}06129342\mathrm{.}79571421\mathrm{.}5471917$

$[19]1\mathrm{.}71891664\mathrm{.}49500372\mathrm{.}05322882\mathrm{.}19757962\mathrm{.}72513832\mathrm{.}59619382\mathrm{.}41715642\mathrm{.}83359522\mathrm{.}7356378$

$[28]2\mathrm{.}80236730\mathrm{.}39854581\mathrm{.}72894433\mathrm{.}29229381\mathrm{.}59156481\mathrm{.}13920553\mathrm{.}60339572\mathrm{.}47893701\mathrm{.}9883181$

$[37]1\mathrm{.}77380962\mathrm{.}50463671\mathrm{.}71564082\mathrm{.}37622601\mathrm{.}44976696\mathrm{.}00458472\mathrm{.}05635841\mathrm{.}06193692\mathrm{.}3455690$

$[46]3\mathrm{.}35283892\mathrm{.}61125861\mathrm{.}69763323\mathrm{.}69139262\mathrm{.}56140880\mathrm{.}67302061\mathrm{.}23734362\mathrm{.}30999251\mathrm{.}2520309$

$[55]2\mathrm{.}20753273\mathrm{.}07231451\mathrm{.}68016163\mathrm{.}58954132\mathrm{.}28877232\mathrm{.}26322932\mathrm{.}26701301\mathrm{.}19303912\mathrm{.}7955786$

$[64]0\mathrm{.}45873900\mathrm{.}70224960\mathrm{.}70203031\mathrm{.}18402951\mathrm{.}21611462\mathrm{.}20502792\mathrm{.}14945022\mathrm{.}86566272\mathrm{.}8459733$

$[73]3\mathrm{.}56100092\mathrm{.}75821471\mathrm{.}37843061\mathrm{.}09204621\mathrm{.}52723180\mathrm{.}45973611\mathrm{.}26427231\mathrm{.}79389371\mathrm{.}6919857$

$[82]1\mathrm{.}03936404\mathrm{.}74271733\mathrm{.}04720411\mathrm{.}85129502\mathrm{.}68277152\mathrm{.}94206902\mathrm{.}49477782\mathrm{.}70490352\mathrm{.}7725385$

$[91]0\mathrm{.}15164312$

Consider a random variable, $x,$ following a modified exponential distribution with Cumulative Distribution Function $($CDF$)$:

$F(x)=\{\begin{array}{cc}0,& x<mn>0</mn>\\ 1-\text{e}\text{x}\text{p}& (-x^2),& x0\end{array}$

$($i$)$ Plot the $CDFF(x)$ as a function of $x$ for $x=0,0\mathrm{.}1,0\mathrm{.}2,$dots,$9\mathrm{.}9,10$ when $=0\mathrm{.}2.$

$($ii$)$ Plot the density function $f(x)$ of $x$ as a function of $x$ for $x=0,0\mathrm{.}1,0\mathrm{.}2,$dots,$9\mathrm{.}9,10$ when $=0\mathrm{.}2.$

A random sample of $100$ values of $x$ is provided in randomSample. Rdata. Loading the sample data into $R$ will generate a vector $x$ with $100$ values representing the sample.

$($iii$)$ Calculate the value of the log likelihood function for the parameter $$ at the point $=0\mathrm{.}2$ based on this random sample.

$($iv$)$ Plot the values of the log likelihood function for the parameter $$ based on the sample in randomsample. Rdata. Your plot of the log likelihood function must be for values of $=0\mathrm{.}01,0\mathrm{.}02,$dots,$0\mathrm{.}99,1.$

The maximum likelihood estimator for the parameter $$ based on a random sample $x_1,$dots,$x_N$ is given by:

hat$()=\frac{N}{{\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i^2}$

$($v$)$ Estimate the value of $$ using the maximum likelihood estimator given above and the sample in randomSample. Rdata.

$3$

$($vi$)$ Comment on the plot in part $($iv$)$ and the estimate in part $($v$).$

$2$

$[$Total $24]\mathrm{.}35453341\mathrm{.}22468432\mathrm{.}44131422\mathrm{.}36106830\mathrm{.}58041321\mathrm{.}55027212\mathrm{.}80219051\mathrm{.}1400573$

$[100]3\mathrm{.}0470660$