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3 3.3 Repeat the Lagrangian optimization of (Exercise 3.2) subject to the added constraint that there is a variance value, Var(X) = E(X2)(E(X))2 = σ2.

4 3.4 Using the two‐die roll probabilities from (Exercise 2.3) compute the mutual information between the two die using the relative entropy form of the definition. Compare to the pure Shannon definition: MI(X, Y) = H(X) + H(Y) – H(X, Y).

5 3.5 Go to genbank (https://www.ncbi.nlm.nih.gov/genbank) and select the genomes of three medius‐sized bacteria (~1 Mb), where two bacteria are closely related. Using the Python code shown in ssss1, determine their hexamer frequencies (as in Exercise 2.5 with virus genomes). What is the Shannon entropy of the hexamer frequencies for each of the three bacterial genomes? Consider the following three ways to evaluate distances between the genome hexamer‐frequency profiles (denoted Freq(genome1), etc.), try each, and evaluate their performance at revealing the “known” (that two of the bacteria are closely related):distance = Shannon difference = | H(Freq(genome1))−H(Freq(genome2))|.distance = Euclidean distance = d(Freq(genome1),Freq(genome2)).distance = Symmetrized Relative Entropy= [D(Freq(genome1)||Freq(genome2))+D(Freq(genome2)||Freq(genome1))]/2Which distance measure provides the clearest identification of phylogenetic relationship? Typically it should be (iii).