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Unformatted text preview: deviation 12 . Example 4.9.2 Suppose a fair die is rolled 1000 times. What is a rough approximation to the sum of the numbers showing, based on the law of large numbers? 158 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Figure 4.22: Results of simulation of sums of independent random variables, uniformly distributed on [0, 1]. (a) S vs. n. (b) Sn vs. n. n Solution: The sum is S1000 = X1 + X2 + · · · + X1000, where Xk denotes the number showing on the k th roll. Since E [Xk ] = 1+2+3+4+5+6 = 3.5, the expected value of S1000 is (1000)(3.5) = 3500. 6 By the law of large numbers, we expect the sum to be near 3500. Example 4.9.3 Suppose X1 , . . . , X100 are random variables, each with mean µ = 5 and variance σ 2 = 1. Suppose also that |Cov(Xi , Xj )| ≤ 0.1 if i = j ± 1, and Cov(Xi , Xj ) = 0 if |i − j | ≥ 2. Let S100 = X1 + · · · + X100 . (a) Show that Var(S100 ) ≤ 120. (b) Use part (a) and Chebychev’s inequality to find an upper bound on P (| S100 − 5| ≥ 0.5). 100 4.9. LAW OF LARGE NUMBERS AND CENTRAL LIMIT THEOREM Solution:...
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