Independence of random variables

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $I$ be an index set. For each $i \in I$, let $X_i$ be a random variable with values in $(E_i, \mathcal{E}_i)$. The family $(X_i)_{i \in I}$ is independent if for each finite nonempty set $J \subseteq I$ and each choice of $B_j \in \mathcal{E}_j$ for $j \in J$, $$P\left(\bigcap_{j \in J} \{X_j \in B_j\}\right) = \prod_{j \in J} P(X_j \in B_j).$$

Remarks

Equivalently, the σ-algebras $\sigma(X_i) = \{X_i^{-1}(B) : B \in \mathcal{E}_i\}$ are independent. For two random variables this is the independence of two random variables. Mutual independence implies pairwise independence, but the converse is false, as for events. For real random variables $X_1, \dots, X_n$, independence holds if and only if the joint cumulative distribution function is the product of the marginal ones.

Explanations

L1 human

Independent random variables come from separate parts of an experiment that do not influence each other. For $n$ tosses of a coin, the results of the tosses are independent random variables.