Q. 2(Speech model).Markov chains are used to model speech. Consider saying the word "beer." Different people might make the phonemes "b", "ee", "r" longer or shorter, depending on the speaker and where the emphasis is put. To model this, speaking the word "beer" is considered to be a Markov chain on the state spaceD={"b","ee","r","."}with the following transition matrix:

P= 0.9 0.1 0 0

0 0.99 0.01 0

0 0 0.95 0.05

0 0 0 1

1. Draw the state diagram. What states are transient or recurrent?
2. What is the expected amount of time it takes to reach the end of the word "."?
3. Whatistheprobabilitythatafterthreeunitsoftime,youhavenotyetfinishedenun- ciating the first phoneme "b"?