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Pr{yr} ≈ Fr(a)

where FY(a) = PR{ya} is the cdf of intakes evaluated at a, for a = EAR In fact, it is easy to show that when E(r) = E(y):

Pr(yr)= FY(EAR)

The prevalence of inadequate intakes can be assessed as long as one has an estimate of the usual nutrient intake distribution (which is almost always available) and of the median requirement in the population, or EAR, which can be obtained reliably from relatively small experiments.

The quantile FY(EAR) is an approximately unbiased estimator of Pr{yr} if

  • ƒY,R(y,r) = fY(y) fR(r), that is intakes and requirements are independent random variables.

  • Pr{r ≤ –α} = Pr{r ≥ α} for any α > 0, that is, the distribution of requirements is symmetrical around its mean; and

  • > , where and denote the variance of the distribution of requirements and of intakes, respectively.

When any of the conditions above are not satisfied, FY(EAR) ≠ Pr{yr}, in general. Whether FY (EAR) is biased upward or downward depends on factors such as the relative sizes of the mean intake and the EAR.

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