Proof that mutual statistical independence implies pairwise independence

Solution 1:

Mutual independence means the four identities you copied, pairwise independence means the first three of these identities. Ergo.

Solution 2:

For an example where $P[A,B,C]=P[A]P[B]P[C]$ does not imply $P[A,B]=P[A]P[B], P[B,C]=P[B]P[C], P[A,C]=P[A]P[C]$ consider the distribution with indicators:

A  B  C  Prob
1  1  1   1/8
1  1  0   3/8
0  0  1   3/8
0  0  0   1/8

So $P[A]=P[B]=P[C]=\frac12$ and $P[A]P[B]P[C]=\frac18$ but $P[A,B]=\frac12, P[B,C]=\frac18, P[A,C]=\frac18$

In such as case, $A,B,C$ are not mutually independent

Proof that mutual statistical independence implies pairwise independence

Solution 1:

Solution 2:

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