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lim P n→∞ Sn − nµ
√
≤c
nσ 2 = Φ(c). The practical implication of the central limit theorem is the same as that of the DeMoivreLaplace
limit theorem. That is, the CLT gives evidence that the Gaussian approximation, discussed in
Section 3.6.3, is a good one, for sums of independent, identically distributed random variables.
Figure 4.23 illustrates the CLT for the case the Xk ’s are uniformly distributed on the interval [0, 1].
The approximation in this case is so good that some simulation programs generate Gaussian random
variables by generating six uniformly distributed random variables and adding them together to
produce one approximately Gaussian random variable. Example 4.9.5 Let S denote the sum of the numbers showing in 1000 rolls of a fair die. By
the law of large numbers, S is close to 3500 with high probability. Find the number L so that
P {S − 3500 ≤ L} ≈ 0.9. To be deﬁnite, use the continuity correction. 160 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Figure 4.23: Comparison of the pdf of Sn to the Gaussian pdf with the same mean an...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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