Full credit will only be given to correct answers with a clear explanation of how they are obtained. Use additional paper as necessary.

1. The following is the transition probability matrix of a Markov nucleotide sequence. Fill in the blanks
   
   

   P  =	(	0.2	0.3	0.1		)
   		0.4		0.1	0.2
   		0.5	0.1		0.2
   			0.2	0.3	0.4

2. For a Markov chain X₀, X₁, X₂, ... with transition probability matrix P as in question 1, suppose the probability distribution of X₀ is
   
   

   x	fo(x)
   A	1/4
   C	1/4
   G	1/4
   T	1/4

   
   
   That is, the initial nucleotide may be any of the four bases equally likely.
   
   Work out the probability distribution of X₁. (Hint: Use the Law of Total Probability:
   
   P(E) = SUM_i [P( E and B_i )]
                   = SUM_i [P(B_i) * P(E | B_i)].)
   
   

   x	f1(x)
   A
   C
   G
   T

   
   
   
   
   Then also work out the probability distribution of X₂.
   
   

   x	f2(x)
   A
   C
   G
   T

   
   
   Can you suggest a method for finding the probability distribution of X_n?
3. Construct a Markov chain model for a nucleotide sequence generated according to these rules:
   
   

   (i). The present nucleotide is equally likely to be A, C, G, T if the preceding two nucleotides are identical.

   (ii). The present nucleotide will be twice as likely to be C or G than A or T if the preceding two nucleotides are different. Furthermore, when making a choice between C versus G and A versus T, purines will be used 60% of the time.

   
   
   Write out its transition probability matrix.