What is Independent random variables? — Mathematics

Definition 34.2 High School Mathematics · Chapter 34 — Sums of Random Variables and the Law of Large Numbers

$X$ and $Y$ are independent if for all values $x, y$ :

\P(X = x \text{ and } Y = y) = \P(X = x)\,\P(Y = y).

Several variables $X_1, \dots, X_n$ are independent if this product rule holds for every choice of values of every subfamily.

Examples

Example 34.5 (Binomial revisited)

A binomial variable $X \sim \mathcal B(n, p)$ is a sum $X = X_1 + \dots + X_n$ of $n$ independent Bernoulli variables. Hence, structurally:

\E(X) = np, \qquad \V(X) = np(1-p),

recovering Theorem 33.8 without any computation.