0

We know:

Mutual Independence :

For $n‎\geq‎‎‎‎3$, random variables $X_1,X_2,...,X_n$ are mutually independent if $$ p(x_1,x_2,...,x_n)=p(x_1)p(x_2)...p(x_n)$$ for all $x_1,x_2,...,x_n$.

Pairwise Independence :

For $n‎\geq‎‎‎‎3$, random variables $X_1,X_2,...,X_n$ are pairwise independent if $ X_i,X_j$ are independent for all $1\leq i<j\leq n$.

Note that mutual independence implies pairwise independence.(Proof that mutual statistical independence implies pairwise independence) show that the converse is not true.

Personally, I think the answer is cleared with the definition of 'Conditional Independence', but any help is appreciated.

Community
  • 1
s.m.h
  • 33

1 Answers1

3

Easiest example: Suppose three people are each tossing a fair coin. Consider the three events $(A,B)$ match, $(B,C)$ match, and $(A,C)$ match.

Clearly these are pairwise independent. But any two determine the third.

lulu
  • 76,951
  • Sir shouldnt the mutual independence means that all terms should be pairwise indepedend and all simultaneously independent isnt ? If even one fails its not really necessary other three will be true right ? All four must be satified for mutual independence (for three events case ) – ProblemDestroyer Apr 10 '22 at 08:55
  • @ProblemDestroyer Yes. But the problem here asked for an example in which pairwise independence held but mutual independence failed. – lulu Apr 10 '22 at 10:36