Find solution

Let X1, ... . Xn ~ N (H, 62), let Fn (1) = - > n 1=1 denote their empirical distribution, and let @,2 denote the cdf of the distribution N (1, 62). Recall that in the Kolmogorov-Smirnov test, we considered the test statistic In = vnsup | Fn (1) - DH,62 | ER In the Kolmogorov-Lilliefors test, we consider the test statistic Tn = Vnsup |Fx (1) - Qual IER It is true or false that T', and T' , have the same distribution for for all n E N? (Refer to the slides.) O True O FalseA Markov chain with state space {1, 2, 3} has transition probability matrix 0.6 0.3 0.1\\ P. = 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (b) What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (c) Consider a general three-state Markov chain with transition matrix P11 P12 P13 P = P21 P22 P23 P31 P32 P33 Give an example of a specific set of probabilities p;; for which the Markov chain is not irreducible (there is no single right answer to this, of course !).8. (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 M,, is obtained by dividing S, 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 - 51 2 () 56, where e and o are some prespecified tolerances. Determine how the value of n recommended by the Chebyshev inequality changes in the following cases. The best guarantee will be when P(IMn - f1 2 () - Ane? (a) (5 points) The value of e is reduced to half its original value. (b) (5 points) If the error probability o is to be reduced to 6/2, while keeping e the same.1. Suppose Z ~ N(0, 1). Need to find m, and r, such that P(a 1.96) = 0.025 Hence 1 = -1.96 and The = 1.96. 2. If the height of a storm surge following a hurricane has expected value E[X'] = 5.5 feet and the standard deviation ox = 1 foot, use Chebyshev inequality to find and upper bound on P[X > 11]. Solution: We can write P[X > 11] = P[X -/x 2 11 - px] = P[X -/x| >5.5] Using Chebyshev inequaltiy, we get Var[X P[X > 11] 0.033 (5.5)2