3. You know that a certain trait ooours in 30% of the population. You want to know how many individuals you have to draw from the population, on average, to get three Subjects with the trait. A. Describe how you would use a table of random numbers to do a simulation of this situation. [11 points} B. Use the table of random numbers below to carry out five repetitions in the simulation method you described in {A} above. Mark the table so that it's clear to the reader how you did the simulation. Begin with the first element of the first row. It it's necessary to continue on past the rst row, start at the beginning of each row. Based on your ue simulations, what's your best guess as to the number of trials required to achieve three successes? {11 points} 26411 94292 96349 92262 32633 35963 94165 46514 36911 9993? 52195 33966 94331 19656 42211 65491 92313 32563 63494 21329 26556 45934 65431 5933!] Problem 7.4 (10 points) A Markov chain Xo, X1, X2, ... with state space S = {1, 2,3, 4) has the following transition graph: 0.5 0.5 0.5 1 0.5 0.5 0.5 2 0.5 0.5 (a) Provide the transition matrix for the Markov chain. (b) Determine all recurrent and all transient states. (c) Determine all communication classes. Is the Markov chain irreducible? (d) Find the stationary distribution. (e) Can you say something about the limiting distribution of this Markov chain?2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix 4 2 2 Q = 3 1 5 0 -5 (a) What is the expected amount of time spent in each state? (b) What is the transition probability matrix of the embedded discrete-time Markov chain? (c) Is this continuous-time Markov chain irreducible? (d) Compute the stationary distribution for the continuous-time Markov chain and the em- bedded discrete-time Markov chain and compare the two