The following formulas are also provided to save you the trouble of looking them up elsewhere: . Binomial coefficient n! K = k!(n - k)! . Inclusion-exclusion principle P(AUB) = P(A) + P(B) - P(An B) . Chain rule P(AnB) = P(A|B) P(B) . Odds P(A) P(A) Odds(A) = P(A) ~ 1- P(A) . Conditional probability P(A B) = P(An B) P(B) . Bayes' theorem P(E H ) P(Hi) P(HE)= P(E Ho) P(Ho) + P(E|HI) P(H1) + . .. + P(EH,.) P(H,) . Information content I(A) = - 10g2 P(A) = log2(1/P(A)) . Entropy H(X) = -> P(x;) log2 P(xi) i= 13. (24 points) In a study on animal behavior, a newly discovered creature is observed at one- hour intervals. The behavior of the creature is classied into three types: foraging for food (F), resting (R), and traveling (T). The creature's behavior during the daytime is found to be approximately described by a Markov chain with these three states, and the following transition probabilities: o If the creature is foraging one hour, the behavior in the next hour will be foraging 50% of the time, resting 30% of the time, and traveling 20% of the time o If the creature is resting one hour, the behavior in the next hour will be foraging 30% of the time, resting 40% of the time, and traveling 30% of the time o If the creature is traveling one hour, the behavior in the next hour will be foraging 70% of the time, resting 20% of the time, and traveling 10% of the time (a) Sketch a diagram of this Markov chain. (b) Write the transition probabilities in the form of a matrix. (c) Consider the following two state probability vectors: v1 = (033,052,015) v; = (048,031,021) Multiply each of these vectors by the transition matrix. Based on the result, is either one a stationary distribution for this Markov chain? (d) Approximately what percentage of time does this creature spend foraging for food