Ques $2.$ Marks$]$ The figure below shows a sample HMM for the ice cream task. The two hidden states $(H$ and $C)$ correspond to hot and cold weather, and the observations $($drawn from the alphabet $O=\{1,2,3\}$ correspond to the number of ice creams eaten by Jason on a given day.

Assume that $q_t$ is the state on day $t$ and $o_t$ is the observation on day $t.$ Compute the following for the given HMM model:

a$.$ Probability of observation sequence $(3,1,3)$ generated by HMM i$.$e$.,P(o_1=3,o_2=1,o_3=3).$

b$.$ Assuming it is COLD on day $1,$ calculate the probability of observation state as $2$ on day $2$ i$.$e$.,2).$

c$.$ Assuming it is COLD on day $1,$ What is $P(q_2=$COLD$|o_2=2)?$

$($P$.$T$.$O$)$

d$.$ What is COLD, $q_2=$ HOT, $q_3=$ HOT, $q_4=$ HOT, $q_5=$ HOT$)?$