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Unformatted text preview: ition follows from the Chebychev inequality, (2.9), applied to the random
variable Sn .
The law of large numbers is illustrated in Figure 4.22, which was made using a random number
generator on a computer. For each n ≥ 1, Sn is the sum of the ﬁrst n terms of a sequence of
independent random variables, each uniformly distributed on the interval [0, 1]. Figure 4.22(a)
illustrates the statement of the LLN, indicating convergence of the averages, Sn , towards the mean,
0.5, of the individual uniform random variables. The same sequence Sn is shown in Figure 4.22(b),
except the Sn ’s are not divided by n. The sequence of partial sums Sn converges to +∞. The LLN
tells us that the asymptotic slope is equal to 0.5. The sequence Sn is not expected to get closer to
2 as n increases–just to have the same asymptotic slope. In fact, the central limit theorem, given
in the next section, implies that for large n, the diﬀerence Sn − n has approximately the Gaussian
distribution with mean zero, variance 12 , and standard...
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- Spring '08
- The Land