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If I have a discrete random process (✗n) n≥20 on two states where for each n, ✗, is independent of X-3, Xn-4, ... Xo when conditioned on Xn-1 and Xn-2, then a. This process is a Markov chain because the future is independent of the past conditioned on the present. b. This process is not a Markov chain but it can be turned into a Markov chain on 4 states. c. This process is a Markov chain because although it does not fit the exact definition, the way it differs is just a technicality. d. This process is not a Markov chain because it does not fit the mathematical definition and this definition is quite strict. e. This process is not a Markov chain but it can be turned into a Markov chain on 3 states.

Which of the following processes does not have the Markov property? a. The process (✗n) n≥0 of sampling a ball from an urn and noting its colour, if one samples without replacement. b. The process (Xn)n20 of sampling a ball from an urn and noting its colour, if one samples with replacement. c. The process (✗n)n≥0 defined by X = Y1 + ··· + Y2 for n ≥ 1, where for each n, Y equals one if a fair coin toss lands heads, and equals 0 otherwise. d. The process (Xn) n≥0 where X represents the outcome of rolling a fair 6-sided die. e. The process (X)n≥0 where X represents the outcome of rolling a fair 6-sided die for 0 ≤ n ≤ 5 and the outcome of throwing a fair coin for n> 6.

Two machines are running continuously, with their operating states described by two Markov chains. For each machine, the possible states are {1, 2, 3, 4, 5, 6}. The state of the first machine on day n is ✗r. The state of the second machine on day n is Yr. The transition matrices for (Xn)n20 and (Y)n20 are P and Q respectively where: 1/3 2/3 0 0 0 0 01 0 1/3 4/9 2/9 0 0 0 0 000 2/3 1/3 0 0 0 0 0 1/3 0 1/3 1/3 0 0 0 0 1/2 1/2 P = ; Q: = 0 0 0 1/3 4/9 2/9 0 0 0 0 2/3 1/3 2/3 0 0 0 1/3 0 1 0 0 0 0 0 2/3 0 0 00 1/3 1 0 0 00 0 Let w = (1/4 1/4 1/6 1/12 5/36 1/9). Then a. the vector w is a limiting distribution for (✗n)n≥0 and (Yn)n≥0; O b. the vector w is a limiting distribution for (✗n) n≥0 but not (Yn)n>0; ○しろまる c. ○しろまる d. the vector w is a limiting distribution for neither (✗n) n≥0 nor (Yn)n≥0; the vector w is a limiting distribution for (Yn) n≥0 but not (✗n)n≥0;

Consider the Markov chain with incidence matrix (1000) a. Pil b. Pii equals 0 (1000) is nonzero but very small (1000) c. P36 d. P14 is close to 1 (1000) is close to 1 0 1/2 0 1/2 00 0 0 1/3 0 2/3 0 1/4 0 1/2 0 0 1/4 0 0 0 1 0 0 0 0 0 0 1 0 0 0 000 1

Let (Xn)n≥o be a sequence of independent, identically distributed random variables, where each X is the outcome of rolling a biased 4-sided die, taking values in S = {1,2,3,4} and with probability distribution P(X = 1) = 1/2, P(X = 2) = P(X = 3) = P(X = 4) = 1/6. Which of the following is true? a. (Xn) no is not a Markov chain as the outcome of ✗+1 does not depend on the outcome of X. b. (Xn)no is not a Markov chain as the outcome of Xn+1 depends on the outcome of Xo. c. (X)n20 is a Markov chain as the future only depends on the present. d. (Xn)no is a Markov chain because it has an equilibrium distribution μ = (1/2, 1/3, 1/6). e. (Xn)no is a Markov chain because it is a sequence of independent random variables.

24. Education Tax The table shows the results of a survey that asked 506 Maine adults whether they favored or opposed a tax to fund education. A person is selected at random from the sample. Find the probability of each event. (Adapted from Portland Press Herald) Support Oppose Unsure Total Males 128 99 20 247 Females 173 259 Total 301 164 41 506 (a) The person opposes the tax or is female. (b) The person supports the tax or is male. (c) The person is not unsure or is female.

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It is believed that the 10% of the population is left-handed. Find the probability that in a random sample of 200 people, between 15 to 30 people are left-handed. If you were told that in a random sample of 70 people "no more than one person is left-handed", would you believe this claim? Show the mathematical calculations relevant to your answer. HINT: Think about the sampling distribution of sample proportion.

A paint manufacturer claims that his paint will last on average 5 years. Assuming that lifetime of his paints is normally distributed and has a standard deviation of half a year, answer the below questions. Suppose that you are a contractor and you painted one house with this paint. It lasted for 4.5 years. Would you consider this as evidence against manufacturers claim? Show calculations relevant to your answer. Suppose that you paint 15 houses with his paint. It lasted an average of 4.5 years for the 15 houses. Would you consider that as evidence against the manufacturer’s claim? Show calculations relevant to your answer.