#@title Q$3$ Data

data$\_$q$3=$c$(0\mathrm{.}872519197149379,0\mathrm{.}107379609259172,0\mathrm{.}5513882695116,0\mathrm{.}462679539768645,$

$0\mathrm{.}755169234299627,0\mathrm{.}335300886362438,0\mathrm{.}238799817476665,0\mathrm{.}501417518738601,$

$1\mathrm{.}03309716208362,1\mathrm{.}17330806654073,3\mathrm{.}06093852844009,0\mathrm{.}562000746297666,$

$0\mathrm{.}114718391601987,0\mathrm{.}0829799299353635,0\mathrm{.}512723825323994,0\mathrm{.}667636522489739,$

$0\mathrm{.}82129174163406,0\mathrm{.}900174311073836,0\mathrm{.}393283364858806,0\mathrm{.}17233043703901,$

$0\mathrm{.}594318190460774,0\mathrm{.}961380988352977,0\mathrm{.}894979588441959,0\mathrm{.}886134558996956,$

$1\mathrm{.}26220813146941,1\mathrm{.}34489718177054,0\mathrm{.}596352108849055,0\mathrm{.}339763989260709,$

$0\mathrm{.}747981143716208,1\mathrm{.}35320439789864,0\mathrm{.}784479379144194,1\mathrm{.}32393253970876,$

$0\mathrm{.}100060523776539,1\mathrm{.}27031586844816,0\mathrm{.}504314863802574,0\mathrm{.}56460463732528,$

$0\mathrm{.}169277806640116,0\mathrm{.}233277206123823,0\mathrm{.}029565847209093,0\mathrm{.}389537682045547,$

$0\mathrm{.}377710850624072,0\mathrm{.}498747425254129,0\mathrm{.}751633832498958,0\mathrm{.}230597467466196,$

$0\mathrm{.}644018423459058,1\mathrm{.}02302531624599,0\mathrm{.}62188860328933,0\mathrm{.}565494144900288,$

$0\mathrm{.}475945173563433,0\mathrm{.}519465268971357)$ Use the $``$optim$\text{'}\text{'}$ function R to obtain the numerical MLEs. Optimizing over both alpha and beta. Complete the code outline below by filling in the $``...\text{'}\text{'}$ terms. The data is generated from a Gamma$(2,3)$ distribution initial$\_$values$=$c$(1,1)$

optfun$\_2$param$=$function$($params$)\{$

alpha$=$params$[1]$

beta$=$params$[2]$

#your code here:

return$(...)$

$\}$

#fill in elipses

mle$\_$out$2=$optim$(...,...,$lower$=$c$(0,0),$method$=\text{'}$L$-$BFGS$-$B$\text{'},$control$=$list$($fnscale$=-1))$

#print MLEs:

print$($paste$(\text{'}$alpha $=\text{'},$mle$\_$out$2$$par$[1]))$

print$($paste$(\text{'}$beta $=\text{'},$mle$\_$out$2$$par$[2]))$ initial$\_$values$=1$

optfun$\_1$param$=$function$($params$)\{$

alpha$=$params$[1]$

beta$=$#your code here

#your code here:

return$(...)$

$\}$

#fill in elipses:

mle$\_$out$1=$optim$(...,...,$lower$=0,$method$=\text{'}$L$-$BFGS$-$B$\text{'},$control$=$list$($fnscale$=-1))$

#print MLEs:

print$($paste$(\text{'}$alpha $=\text{'},$mle$\_$out$1$$par$[1]))$

#your code here:

print$($paste$(\text{'}$beta $=\text{'},...))$