Question $2$: Consider the following training corpus $\backslash (\backslash$mathbf$\{$T$\}\backslash )$ of sentences:

$-$ START Karlsson is round STOP

$-$ START He lives on the roof STOP

$-$ START He is happy STOP

$-$ START On the roof STOP

$-$ START Karlsson lives happily STOP

$($b$)$ Compute the following maximum likelihood parameters:

$\backslash (\backslash$mathrm$\{$p$\}(\backslash )$ Karlsson $\backslash (\backslash$mid S T A R T$)=\backslash )$

p $($Karlsson$|$lives$,$ happily $\backslash ()=\backslash )$

$\backslash (\backslash$mathrm$\{$p$\}(\backslash )$ STOP$|$happy $\backslash ()=\backslash )$

$($c$)$ Compute the probability of the following sentences under the trigram model trained on $\backslash (\backslash$mathbf$\{$T$\}\backslash )$ :

START Karlsson is happy STOP

START Karlsson lives on the roof STOP