I've just read in a stochastics textbook:

Let $(\Omega, P)$ be a discrete probability space.

(a) The events $A_i \subseteq \Omega, i=1,2, \dots$ are called independent, if $$P(A_{i_1} \cap A_{i_2} \cap \dots \cap A_{i_k}) = P(A_{i_1}) \cdot \dots \cdot A_{i_k}$$ for all finite sets $\{i_1, \dots, i_k\} \subsetneq \{1,2, \dots\}$ and all $k \geq 2$.

(b) The events $A_i \subseteq \Omega, i=1,2,\dots$ are called pairwise independent, if $$P(A_{i_1} \cap A_{i_2}) = P(A_{i_1}) \cdot P(A_{i_2})$$ for all pairs $\{i_1, i_2\} \subsetneq \{1,2,\dots\}$.

After these two definitions it states that pairwise independent events are not always independent.

Do you have an example?

  • $\begingroup$ Please note that English is not my mother tongue and the textbook is written in German. Please correct my mistakes if I used the wrong terms (e.g. I'm not sure if it is called "event" in English). $\endgroup$ – Martin Thoma Feb 11 '15 at 8:20
  • $\begingroup$ German Wikipedia has this example. $\endgroup$ – Brian M. Scott Feb 11 '15 at 8:24
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    $\begingroup$ My favourite formulation of Michael's standard example is to let $X$ be a uniformly random angle and take the events $\sin(X)>0$, $\cos(X)>0$, $\tan(X)>0$. $\endgroup$ – Erick Wong Feb 11 '15 at 8:37
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    $\begingroup$ My favourite formulation of Michael's standard example (which has been posted maaany times on the site) is to let $X_1=U$, $X_2=V$, $X_3=UV$, with $U$ and $V$ i.i.d. uniform on $\{-1,+1\}$. This generalizes to higher dimensions. $\endgroup$ – Did Feb 11 '15 at 8:40

(I am just rewriting my comment above in the form of an answer).

A standard example is to define $X_1, X_2$ as independent Bernoulli random variables with $Pr[X_i=1]=Pr[X_i=0]=1/2$ for $i \in \{1,2\}$, and define $X_3 = X_1 + X_2 \mod 2$. Then $X_3$ is dependent on $\{X_1,X_2\}$, so $\{X_1,X_2,X_3\}$ are not mutually independent. However, any two out of the three random variables are independent. That is, $\{X_1, X_2\}$ are independent, $\{X_2,X_3\}$ are independent, and $\{X_1,X_3\}$ are independent.


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