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2. (10 points) (The Weak Law of Large Numbers) In order to estimate f, the true fraction of smokers in a large population, Alvin selects n people at random. His estimator Mn is obtained by dividing Sn, the number of smokers in his sample, by N, i.e., Mn = Sn. Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that P(IMn - f1 2 6) 58, where e and o are some prespecified tolerances. Determine how the value of n recommended by the Chebyshev inequality changes in the following cases. (a) (5 points) The value of e is reduced to half its original value. (b) (5 points) The value of o is reduced to half its original value.Q.4 [12 marks] Give examples for Markov chains with the following properties. (a) The Markov chain has a state which is both transient and essential. [3 marks] (b) The Markov chain has a state which is null recurrent and aperiodic. [3 marks (c) The Markov chain has a stationary but not a limiting distribution? [3 marks] (d) The Markov chain has a stationary distribution which is not unique. 3 marks]What is the difference between univariate data and bivariate data? Choose the correct answer below. O A. In univariate data, a single variable is measured on each individual. In bivariate data, two variables are measured on each individual. O B. In univariate data, there are only positive values and zeros. In bivariate data, there are positive values, negative values, and zeros. C. In univariate data, there is one mean. In bivariate data, there are two means. O D. In univariate data, the data are qualitative. In bivariate data, the data are quantitative.Problem 7.4 (10 points) A Markov chain X0,X1,X2,. . . with state space S = {1,21 3,4} has the following transition graph: (3) Provide the transition matrix for the Markov chain. (b) Determine all recurrent and all transient states