For the following Markov models a) draw the transition diagram, if one is not provided; b) put the states in (some) order and write down the transition matrix; c) calculate the probability of the given strings of states, taking the first state as given (e.g. for a string of 3 states, there are only 2 transitions.)

Use the transition diagram in figure a (Model 1) to calculate the probability of the string of states BAB.

Use the transition diagram in figure b (Model 2) to calculate the probability of the string of states CCD.

Use the transition diagram in figure c (Model 3) to calculate the probability of the string of states EEF.

Use the transition diagram in figure d (Model 4) to calculate the probability of the string of states GGG.

An ion channel can be in either open (O) or closed (C) states. If it is open, then it has probability 0.1 of closing in 1 microsecond; if closed, it has probability 0.3 of opening in 1 microsecond. Calculate the probability of the ion channel going through the following sequence of states: COO.

An individual can be either susceptible (S) or infected (I), the probability of infection for a susceptible person is 0.05 per day, and the probability an infected person becoming susceptible is 0.12 per day. Calculate the probability of a person going through the following string of states: SISI.

The genotype of an organism can be either normal (wild type, W) or mutant (M). Each generation, a wild type individual has probability 0.03 of having a mutant offspring, and a mutant has probability 0.005 of having a wild type offspring. Calculate the probability of a string of the following genotypes in successive generations: WWWW.

There are three kinds of vegetation in an ecosystem: grass (G), shrubs (S), and trees (T) . Every year, 25% of grassland plots are converted to shrubs, 20% of shrub plots are converted to trees, 8% of trees are converted to shrubs, and 1% of trees are converted to grass; the other transition probabilities are 0. Calculate the probability of a plot of land have the following succession of vegetation from year to year: GSGG.

The nAChR ion channel can be in one of three states: resting (R), closed with Ach bound (C), and open (O) with transition probabilities (per one microsecond): 0.04 (from R to C), 0.07 (from C to R), 0.12 (from C to O) and 0.02 (from O to C); the other transition probabilities are 0. Calculate the probability of the following string of states: OCCR.

(Challenging) We considered a sequence of Bernoulli trials in chapter 4, for example a string of coin tosses where each time heads and tails come up with probability 0.5. Describe this experiment as a Markov model, draw its transition diagram and write its transition matrix.

(Challenging) Now do the same for a sequence of Bernoulli trials where success has probability 0.9 (and failure has probability 0.1).

(Challenging) Can you formulate a test, based on a transition matrix of a Markov model, to tell whether it's generating a string of independent random variables as opposed to a string of random variables that depend on the previous one?

Example 1.8. Branching processes. These processes arose from Francis Galton's statistical investigation of the extinction of family names. Consider a population in which each individual in the nth generation independently gives birth, producing k children (who are members of generation n + 1) with proba- bility px. In Galton's application only male children count since only they carry on the family name. To define the Markov chain, note that the number of individuals in genera- tion n, Xn, can be any nonnegative integer, so the state space is {0, 1, 2, . . .}. If we let Y1, Y2, . .. be independent random variables with P(Ym = k) = Pk, then we can write the transition probability as p(i, j) = P(Y1 + . . . + Yi=j) for i > 0 and j 2 0 When there are no living members of the population, no new ones can be born, so p(0, 0) = 1. Galton's question, originally posed in the Educational Times of 1873, is Q. What is the probability that the line of a man becomes extinct?, i.e., the branching process becomes absorbed at 0? Reverend Henry William Watson replied with a solution. Together, they then wrote an 1874 paper entitled On the probability of extinction of families. For this reason, these chains are often called Galton-Watson processes.1. Fill in the details for the proof of the Cauchy-Schweitz inequality for random variables EXY 5 VIEXZEYZ. 2. Show that the \"main" theorem on Galton-Watson processes cannot be extended to cases where the offspring disnibution changes over time. That is, give random variables Xle , . .. each of mean EX,- 2 2 but such that if individuals in generation I each have a number of children EX} (independently of one another} then PtUdZ: =0})=1. 3. Show that, for every tr > 1 and I] 1. Let T denote the correSponding Gabon-Watson tree. and let '1\" denote (the distribution of] T conditioned on extinction. Show that T' is distributed like a Salton-Watson tree with offspring distribution X\" d:P0l:q.u} where q is the extintion probability of T. 1. Each box of a particular brand of cereals contains a plastic figure. There are four types of figures, each equally likely. Let T denote the number of boxes you need to buy until you have all four kinds of figures. Determine the PGF Gr(s) and P(T = k) for each k. 2. X has the Poisson distribution with parameter Y, where Y is itself random. Namely, Y Po(u). Show that Gx+Y(s) = explu(sex-1 -1)]. 3. Consider a Galton-Watson process with offspring distribution geometric from zero (P(X = k) = (1 -p)*p.) Let 7 := min {t : Z, =0} if this minimum exists, and 7 =co otherwise. Determine P(T = (). 4. Consider a Galton-Watson process with offspring distribution X " Bi(2, p). (i) Determine the extinction probability q explicitly; (ii) Determine Var(Z, ). (iii) Show that when p = ; then P(Z > 0) ~ 8/1. (Here ~ means the ratio tends to one.)c. If I weighed 15 pounds more than I do, what percentile would I be in? A production process manufactures items with weights that are normally distributed with mean 10 pounds and standard deviation 0.1. An item is considered to be defective if its weight is less than 9.8 pounds or greater than 10.2 pounds. Suppose that these items are currently produced in batches of 1000 units. a. Find the probability that at most 5% of the items in a given batch will be defective b. Find the probability that at least 85% of these items in a given batch will be acceptable. O Mathematical Statistics Probability and Solutions Probability and textbook with Applications Statistics for Engineers Statistics for 7th Edition 9th Edition Oth Edition