Independence of two random variables

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X$ be a random variable with values in $(E_1, \mathcal{E}_1)$ and $Y$ a random variable with values in $(E_2, \mathcal{E}_2)$. $X$ and $Y$ are independent if for all $A \in \mathcal{E}_1$ and $B \in \mathcal{E}_2$, $$P(X \in A, \, Y \in B) = P(X \in A) \, P(Y \in B).$$

Remarks

In other words, the events $\{X \in A\}$ and $\{Y \in B\}$ are independent for all $A$ and $B$. For real random variables, it is enough to check $P(X \le x, Y \le y) = P(X \le x) \, P(Y \le y)$ for all real $x$ and $y$. For discrete random variables, independence is equivalent to $P(X = x, Y = y) = P(X = x) \, P(Y = y)$ for all $x$ and $y$. If $X$ and $Y$ are independent and $g$, $h$ are measurable functions, then $g(X)$ and $h(Y)$ are independent.

Explanations

L1 human

Two random variables are independent when knowing the value of one tells you nothing about the other. The results of two separate dice are independent. The result of one die and the sum of both dice are not.