Definition · applies to events and random variables · also called independent, independence, independent random variables, independent events

High school

Consider two events, such as “the first coin lands heads” and “the second coin lands heads.” The events are independent if the probability of each event is the same whether or not the other event occurs.

Undergraduate (events)

Consider a probability space $(\Omega, \mathcal{F}, P)$. Two events $A$ and $B$ are independent if

\[P(A \cap B) = P(A) \, P(B).\]
Graduate (random variables)

Two random variables $X$ and $Y$ are independent if the events $\setdef{X \in A}$ and $\setdef{Y \in B}$ are independent for every pair of Borel sets $A$ and $B$.

Properties

  • If $P(B) > 0$, then $A$ and $B$ are independent exactly when $P(A \mid B) = P(A)$.
  • If $X$ and $Y$ are independent and have finite variances, then $X$ and $Y$ are uncorrelated, but uncorrelated random variables can be dependent: if $X$ is uniform on $[-1, 1]$ and $Y = X^2$, then $\operatorname{Cov}(X, Y) = 0$.
  • If $X$ and $Y$ are jointly normal, then $X$ and $Y$ are independent exactly when $X$ and $Y$ are uncorrelated.
  • Statistical independence of random variables is different from linear independence of vectors.

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