$($Exercise $3\mathrm{.}4$ & $3\mathrm{.}7$ of chapter $3$ of SLP book$)$ Given the following corpus:

I $am$ Sam

Sam I $am$

Sam I like

Sam I $do$ like

I like Sam $$

$($a$)$ Using a bigram language model with add$-$one smoothing, what is $P(Sam|am)?$

Include in your counts just like any other token

$($b$)$ If we use linear interpolation smoothing between a maximum$-$likelihood bigram

model and a maximum$-$likelihood unigram model with ${}_1=\frac{1}{2}$ and ${}_2=\frac{1}{2},$ what is

$P(Sam|am)?$ Include in your counts just like any other token.

$($c$)$ Which of the following sentences gets a higher probability with this model?

I $am$ Sam

Sam I $am$

$($d$)$ What is the perplexity of this model if evaluated on a toy test dataset that contains

only one sentence?