The acronymi.i.d. is particularly common in statistics, where observations in a sample are typically assumed to be (more-or-less) i.i.d. for the purposes of statistical inference. The assumption (or requirement) that observations be i.i.d. tends to simplify the underlying mathematics of many statistical methods.
Examples
The following are examples or applications of independent and identically distributed (i.i.d.) random variables:
All other things being equal, a sequence of outcomes of spins of a roulette wheel is i.i.d. From a practical point of view, an important implication of this is that if the roulette ball lands on 'red', for example, 20 times in a row, the next spin is no more or less likely to be 'black' than on any other spin.
One of the simplest statistical tests, the z-test, is used to test hypotheses about means of random variables. When using the z-test, one assumes (requires) that all observations are i.i.d. in order to satisfy the conditions of the central limit theorem.
Although this region is a desert, with high temperatures and low rainfall of three inches (75 mm) per year, the economy is heavily based on agriculture due to the availability of irrigation water, which is supplied wholly from the Colorado River via the All-American Canal.
A vast system of canals, check dams, and pipelines carry the water all over the valley, a system which forms the Imperial Irrigation District, or IID.
Many visitors come to the area to visit the Salton Sea (California's largest inland lake, which serves as a dumpout point for the overflow and drainage from the IID canal system and ditch drainage) and the Algodones Dunes.
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