Independent and identically distributed random variables

Definition

Let $(X_i)_{i \in I}$ be a family of random variables on one probability space, all with values in the same measurable space $(E, \mathcal{E})$. The family is independent and identically distributed (i.i.d.) if both of these conditions hold:

1. The family $(X_i)_{i \in I}$ is independent.
2. The random variables $X_i$ are identically distributed: all $X_i$ have the same law $\mu$.

Remarks

One also says that $X_1, X_2, \dots$ is an i.i.d. sequence with common law $\mu$, or a random sample from $\mu$. The results of repeated tosses of one coin form an i.i.d. sequence of Bernoulli random variables. The law of large numbers and the central limit theorem are first proved for i.i.d. sequences.

Explanations

L1 human

I.i.d. variables are the results of the same experiment, repeated under the same conditions, with no influence between the repetitions. Each one follows the same law, and none of them gives information about the others.