Independent and Identical Distributed Samples

Let X = (X1, . . . , Xn) where X1, . . . , Xn are random variables. We call X a random vector. Let f(x1, . . . , xn) denote the pdf.

If X1, . . . , Xn are independent and each has same marginal distribution with density f, we say that X1, . . . , Xn are IID.

To get a better feeling of IIDs, let us take an example.
In Casino games, the structure yields independent and identically distributed (IID) outcomes. Each iteration of a roll of dice, a shuffle of cards or roulette wheel is independent of any other iteration. The probability of a particular outcome is the same at every iteration.

All random variables in an IID share the same probability distribution. It means that if you plot all variables together, they would resemble some kind of distribution. This yields to the fact that each random variable has the same mean and variance.

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