i.i.d. sequence

A sequence of i.i.d. random variables is a sequence $(X_n)_{n\ge 1}$ of random variables on a common probability space such that the family $(X_n)_{n\ge 1}$ is independent and the variables are identically distributed (equivalently, each $X_n$ has the same distribution law as $X_1$).

i.i.d. sequences formalize repeated sampling and are the basic setting for the weak law of large numbers , the strong law of large numbers , and the central limit theorem .

Examples:

• Let $X_n$ be the indicator that the $n$th fair coin toss is heads. Then $(X_n)_{n\ge 1}$ is i.i.d. Bernoulli$(1/2)$.
• Let $X_1,X_2,\dots$ be independent samples from a normal distribution $N(0,1)$. Then $(X_n)_{n\ge 1}$ is an i.i.d. sequence with that common law.