The weak law of large

Markov inequality For a random variable X 2 0 with mean / > 0, and any numbert > 0: P (X > D). Note that the Markov inequality is restricted to non-negative random variables. Chebyshev inequality For a random variable X with (finite) mean / and variance of, and for any number t 2 0, P(X -/| 20) . Remark: When Markov inequality is applied to (X - ()", we obtain Chebyshev's inequality. Markov inequality is also used in the proof of Hoeffding's inequality. Hoeffding versus Chebyshev 4 points possible (graded) Let X1, X2, ..., X'n " Unif (0, b) be ni.i.d. uniform random variables on the interval [0, b] for some positive b. Suppose n is small (i.e. 7 1 ) 0 there holds P(1X - #| Ska) 21 - 1 That is. with probability at least 1 - 2. X stays within & standard deviations around its mean.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)2SECTION 5.2. The Weak Law of Large Numbers Problem 4. 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 - 1 Z E) 56, where e and o are some prespecified tolerances. Determine how the value of n recom- mended by the Chebyshev inequality changes in the following cases. (a) The value of e is reduced to half its original value. (b) The probability o is reduced to half its original value