# Statistics for Environmental Engineers

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A slightly different kind of simulation is bootstrapping. The bootstrap is an elegant idea. Because sampling distributions for statistics are based on repeated samples with replacement (resamples), we can use the computer to simulate repeated sampling. The statistic of interest is calculated for each resample to construct a simulated distribution that approximates the true sampling distribution of the statistic. The approximation improves as the number of simulated estimates increases.

### Monte Carlo Simulation

Monte Carlo simulation is a way of experimenting with a computer to study complex situations. The method consists of sampling to create many data sets that are analyzed to learn how a statistical method performs.

Suppose that the model of a system is y = f (x). It is easy to discover how variability in x translates into variability in y by putting different values of x into the model and calculating the corresponding values of y. The values for x can be defined as a probability density function. This process is repeated through many trials (1000 to 10,000) until the distribution of y values becomes clear.

It is easy to compute uniform and normal random variates directly. The values generated from good commercial software are actually pseudorandom because they are derived from a mathematical formula, but they have statistical properties that cannot be distinguished from those of true random numbers. We will assume such a random number generating program is available.

To obtain a random value YU(a, в) from a uniform distribution over the interval (а,в) from a random uniform variate RU over the interval (0, 1), this transformation is applied:

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