How do I test the independence of two events?

Question

Two events $A$ and $B$ are independent if :

$P(A|B) = P(A)$

However, I was wondering if there was an intuitive way to test for independence - rather than having to check this condition every time.

Answer 1

Two events $A$ and $B$ are independent if and only if $P(A \cap B) = P(A)P(B)$.

Two random variables $X,Y$ with joint distribution function $F_{XY}$ are independent if and only if $F_{XY}$ can be written as

$$F_{XY}(x,y) = P(X \leq x, Y \leq y) = P(X \leq x)P(Y \leq y)$$

for all $x,y$.

Two random variables $X,Y$ with joint density function $f_{XY}$ are independent if and only if there exists some functions $g(x)$ and $h(y)$ such that

$$f_{XY}(x,y) = g(x)h(y)$$

for all $x,y$.

It doesn't really get any simpler than those conditions other than special cases.

One special case is for normally distributed random variables: If $X$ and $Y$ are normally distributed and $E[XY]=E[X]E[Y]$, then $X,Y$ are independent. Note this only holds for normally distributed random variables.

Answer 2

Not really.   That is basically exactly what independence of events means.$$\begin{split}\mathsf P(A\cap B)&=\mathsf P(A\mid B)~\mathsf P(B)&\qquad&\text{by definition}\\ &=\hspace{3.75ex}\mathsf P(A)~\mathsf P(B)&&\text{iff independent}\end{split}$$

Now sometimes it may seem obvious whether independence will turn out to be the case before calculations are begun, but reasonably often such intuition misleads.