Download PDF by Daniel J. Velleman: American Mathematical Monthly, volume 116, number 1, january

By Daniel J. Velleman

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Extra info for American Mathematical Monthly, volume 116, number 1, january 2009

Example text

22) Now demonstrating Eh(Wn ) → Eh(Z) can be accomplished by showing lim E f (Wn ) − Wn f (Wn ) = 0. n→∞ It is easy to verify that when Eh(Z) exists an explicit solution to (22) is given by f (w) = ϕ −1 (w) w −∞ [h(u) − Eh(Z)]ϕ(u)du, (23) where ϕ(u) is the standard normal density given in (2). Stein [11] showed that when h is a bounded differentiable function with bounded derivative h , the solution f is twice differentiable and satisfies || f || ≤ 2||h|| and || f || ≤ 2||h ||, (24) where for any function g, ||g|| = sup −∞

The vi forces left behind tend to be few and far apart. When this process is applied, for example, on the optimal loaded spinal stack of weight 100, only one such external force is needed, as shown in the second diagram of Figure 14. The nonzero vi ’s can be easily realized by erecting appropriate towers, as shown at the bottom of Figure 14. The top part of the balancing set is then designed by solving a small linear program. We omit the fairly straightforward details. 23897 overhang of the non-spinal loaded stack of weight 100 given in Figure 6.

Only many years later with the work of Laplace around 1820 did it begin to be systematically realized that the same normal limit is obtained when the underlying Bernoulli variables are replaced by any variables with a finite variance. The result was the classical Central Limit Theorem, which states that Wn converges in distribution to Z whenever √ Wn = (Sn − nμ)/ nσ 2 is the standardization of a sum Sn , as in (1), of independent and identically distributed random variables each with mean μ and variance σ 2 .

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American Mathematical Monthly, volume 116, number 1, january 2009 by Daniel J. Velleman


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