Let $\{x_t{\}}_{t}0$ be a continuous$-$time Markov chain with state space $\{1,2,3\}$ and rates $q(1,2)=$

$1,q(2,1)=4,q(2,3)=1,q(3,2)=4,q(1,3)=0,$ and $q(3,1)=0.$ Solve the following:

$($a$)$ Determine the transition matrix of the embedded Markov chain $\{Y_n{\}}_{n}0$; i$.$e$.,$ the chain

recording all the consecutive different states of $\{x_t{\}}_{t}0.$

$($b$)$ Find the generator matrix $Q.$

$($c$)$ Below, we give two $R$ codes: One to generate a path $Y_0,Y_1,Y_2,$dots of a Discrete$-$Time

MC $\{Y_n{\}}_{n}0$ and another to generate the arrival times $T_1,T_2,$dots of a Poisson process $N$

with rate $($plus $T_0=0).$ Combine them to write a code to plot one path or trajectory

of the chain $\{x_t\}$ for $t$ in $0,30,$ assuming that the chain starts in state $1($to this end,

it may be useful to use the function stepfun$()$ of R$).$