For \(n \ge 3\), random variables \(X_1, X_2, \cdots, X_n\) are mutually independent if

\[ \begin{aligned} p(x_1, x_2, \cdots, x_n) &= p(x_1) p(x_2) \cdots p(x_n) \\ \end{aligned} \]

and are pairwise independent if \(X_i\) and \(X_j\) are independent, ie

\[ \begin{aligned} p(x_i, x_j) &= p(x_i) p(x_j) \\ \end{aligned} \]

for all \(1 \le i < j \le n\).

What would be an example where pairwise independence holds but not mutual independence ? Here is an idea. Imagine there is a slot machine that has 4 equally likely outcomes:

\[ \begin{aligned} (1,2), (1,3), (2,3), \text{FAIL} \end{aligned} \]

So, for example,

\[ \begin{aligned} p(1) &= p(2) = (1+1)/4 = 1/2 \\ p(1,2) &= p(1) p(2) = 1/4 & \text{pairwise independent} \\ p(1,2,3) &= 0 \not = p(1) p(2) p(3) & \text{not mutually independent} \\ \end{aligned} \]

\(\Box\)

Source: Information Theory.

# posted by rot13(Unafba Pune) @ 4:14 PM