Independent and Identically Distributed (IID)

IID (aka iid, i.i.d.) is a commonly used assumption in statistical modeling, which simplifies the underlying mathematics of a complex system.

When we have a set or sequence of i.i.d. random variables, it means:

• Independent: Each random variable is independent of each other

• Identically Distributed: Each random variable has the same probability distribution

  Do not mistake identical for uniform distribution.

If we have IID random variables $X_1,\dots, X_n$, each with the same CDF $F$, we denote:

$$ X_1,\dots, X_n \sim F $$

Additionally, the random variables $X_1, \dots, X_n$ are called random sample of size $n$ from $F$.