Statistics for Environmental Engineers

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Histogram of 4000 lognormally distributed values

(14 values > 100)

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Histogram of 1000 4-day averages of lognormally distributed values

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FIGURE 50.1 Left-hand panel: frequency distribution of 4000 daily observations that are random, independent, and have a lognormal distribution x = exp( y), where y is normally distributed with n = 2 and с = 1. Right-hand panel: frequency distribution of 1000 4-day averages, each computed from four random values sampled from the lognormal distribution.

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FIGURE 50.2 Distribution of 100 nonparametric estimates and 100 parametric estimates of the 99th percentile, each computed using a sample of n = 100 from the lognormal distribution shown in the top panel.

“violate” the 18-ug/L limit? Will the number of violations depend on whether the 99th percentile is estimated parametrically or nonparametrically? (These two estimation methods are explained in Chapter 8.) These questions can be answered by simulation, as follows.

1.    Generate a set of n = 100 observations from the “true” lognormal distribution.

2.    Use these 100 observations to estimate the 99th percentile parametrically and nonparametrically.

3.    Repeat steps 1 and 2 many times to generate an empirical distribution of 99th percentile values.

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