Question $4.$

Let $x$ be a random variable whose density is given by $f(x)=\{\begin{array}{cc}{e}^{-x},& x>0\\ 0,& x0\end{array},$ where $>0$ is a fixed parameter.

Part $($a$)$

Create $R$ function likely thet $a,x)$ which, for a given value of parameter theta and a given vector $x($which represents a sample from the

population $x),$ returns the corresponding value of the likelihood function of parameter $$ of population $x,$ based on the sample $x.$

Note that since $f(t)=0$ for $t0,$ we can assume that all the entries of the sample are positive. So$,$ think of $R$ variable $x$ as a vector

$(x_1,x_2,$dots,$x_n)$ with $x_i>0,$ for all iin$\{1,2,$dots,$n\}.$

In other words, you don't need to worry about how to handle situation in which $x_{i}0$ for some iin$\{1,2,$dots,$n\}($although in that case the likelihood

equals $0,$ so it's trivial$).$

Your function likely$()$ should be flexible allowing vector $x$ to be of any length.

Hint: After figuring out the mathematical formula for the likelihood function $L($;vec$(x)),$ your $R$ function likely $()$ should have one or two lines of code. No

loops, nor if$-$else statments, please! Loops are okay only when you cannot vectorize.

Make sure that your $($R$)$ code passes ALL of the test provided by the following code:

## check the output of likely$()$ for some combinations of $x$ and theta

theta$1=1\mathrm{.}8$;$x1=c(1\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}15,0\mathrm{.}8,0\mathrm{.}4,0\mathrm{.}06,0\mathrm{.}7)$

theta$2=3\mathrm{.}5$;$x2=c(0\mathrm{.}3,0\mathrm{.}6,0\mathrm{.}05,0\mathrm{.}2,0\mathrm{.}12,0\mathrm{.}77)$

if $($test$\_$that$($desc$="",$

expect$\_$equal$($ abs$($likely$($theta$1,$ x$1)-0\mathrm{.}0448921045151155)<mtext\; id="EditorElement\_1220660"></mtext><mn\; id="EditorElement\_1220661">1</mn>$e$-7,$ TRUE$)$

$\})!=$ TRUE$)$ stop$("$Sorry$,$ wrong answer."$)$

if $($test$\_$that$($desc$="",$ code

expect$\_$equal$($ abs$($likely$($theta$2,$ x$2)-1\mathrm{.}45728892808099)<mtext\; id="EditorElement\_1220801"></mtext><mn\; id="EditorElement\_1220802">1</mn>$e$-7,$ TRUE$)$

$\})!=$ TRUE$)$ stop$("$Sorry$,$ wrong answer."$)$