In probability theory and statistics , a set of random variables is said to be independent (and) equally distributed , if each of them has the same distribution as the others, and all quantities are independent in the aggregate. The phrase "independent identically distributed" is often abbreviated by the abbreviation iid (from the English. Independent and identically-distributed ), sometimes - "n.d."
Applications
The assumption that random variables are independent and equally distributed is widely used in probability theory and statistics, since it allows one to greatly simplify theoretical calculations and prove interesting results.
One of the key theorems of probability theory - the central limit theorem - states that if Is a sequence of independent identically distributed random variables with finite dispersion, then, when to infinity, the distribution of their average is a random variable converges to a normal distribution .
In statistics, it is usually assumed that a statistical sample is a sequence of iid realizations of some random variable (such a sample is called simple ).