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5 Some quantitative attributes arise from combinations of normally—and not necessarily independently—distributed variables. Consider, for example, the weight of an individual whose height is drawn from a normal distribution. It is likely related to some multiplicative combination of height and girth, which will not be normally distributed. While acknowledging such possibilities, the resulting variables may still be "closely enough" to normally distributed as to make the approximation reasonable.

Technically speaking, in order to find the effects of a change in N on expected welfare, one would need to consider the derivative of the above expression with respect to N. The problem can be made simpler, however, by taking a reasonably close approximation. This approximation is derived by ignoring Jensen's inequality6 substituting the "function of the expectation" for the "expectation of the function," i. e.,

Note that I have stated the expectation of welfare conditioned on the size of the set from which samples are drawn. This arises because the greatest order statistic is also stated as a function conditioned on N, E(6\N). Differentiating with respect to TV, I have where the second equality results because price is equal to marginal cost in competitive equilibrium.

Tractable results are further facilitated by taking advantage of another approximation. It is intuitively straightforward that the expectation of the greatest order statistic in a sample of size N is approximately that value of the random variable for which a fraction N/(N + 1) of the sample is less than that value (a formal proof, including bounds on the approximation, is presented in David, 1981). Thus,

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