Please provide R code where relevant. Consider a $3-$state HMM that has observations from gamma distributions $($gamma$-$HMM$).$ A

gamma distribution has probability density function

$f(x)=\frac{{}^{}}{()}{x}^{-1}{e}^{-x},$ for $x>0,$

where $>0$ is the shape parameter and $>0$ is the rate parameter. The file Assign$4$gammahmm.txt

on the resources webpage contains a series of data that can be modelled using a $3-$state gamma$-$

HMM. Download the file to a directory on your computer, open RStudio, and call

x dget$("$C:$/.../$Assign$4$gammahmm.txt$")$

to load the data and save it in the vector $x$ in $R,$ where $C\frac{\text{:}}{d}$ots represents the folder location of

your file.

$($a$)$ Write down the state$-$dependent distribution in each state. That is$,$ write down the expres$-$

sions of $f(x_t|C_t=1),f(x_t|C_t=2),$ and $f(x_t|C_t=3).$

$[3$ marks$]$

$($b$)$ Fit a $3-$state gamma$-$HMM to the data in $x$ using the function HMMest in R$.$ Report the

parameter estimates.

$[2$ marks$]$

$($c$)$ For the model in $($b$),$ use the function Viterbi in $R$ to find the Viterbi path, i$.$e$.,$ the most

likely sequence of states. Plot the classified data using the Viterbi path. That is$,$ plot the

data and use different colors to represent different states. Attach a copy to your assignment.

$[3$ marks$]$