$1-$ Compute the bigram count table, C$($w$2|$w$1)$ for the following: "A skunk sat on a stump. The stump thunk the skunk stunk, the skunk thunk the stump stunk." Put w$1$ in the left hand column, and w$2$ in the top row.$$ Include punctuation, clitics, and sentence start and end markers as individual tokens.$$

$2-$ Compute the bigram probability table, P$($w$2|$w$1)$ for the phrase $$A skunk sat on a stump.$,$ assuming the following overall unigram counts:

C$($a$)=80,$ C$($skunk$)=100,$ C$($sat$)=20,$ C$($on$)=25,$ C$($stump$)=60,$ C$($the$)=90,$ C$($thunk$)=50,$ C$($stunk$)=75,$ C$(,)=80.$Assume there are $50$ sentences in the corpus, and they all end with a period.

$3-$ Compute the probability and perplexity of the sentence in Question $(1)$ using the bigram approximation.$$

$$Smoothing$$

$4-$ Smooth the count table you calculated in Question $(1)$ using Laplace smoothing, and recalculate the probability table as well. Assume V$=30.$

$5-$Recalculate the probability and perplexity of the first sentence in Question $(1)$ using the smoothed table.