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Suppose we have three random variables $X_1, X_2$ and $X_3$. For pairwise independence it is sufficient to show that $$E[X_{1}X_{2}] =E[X_1]E[X_2]$$, $$E[X_{1}X_{3}] = E[X_1]E[X_3]$$ and $$E[X_{2}X_{3}] = E[X_{2}]E[X_{3}]$$

For mutual independence would it be enough to show that $$E[X_{1}X_{2}X_{3}] = E[X_{1}]E[X_{2}]E[X_{3}]$$?

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Those 3 equalities are not enough to show pairwise independence. For example, take $X_1=0$ and let $X_2$ and $X_3$ be uncorrelated but not independent random variables. – Ben Derrett Mar 12 '12 at 10:41
Also note that the first three conditions you wrote are not sufficient for pairwise independence as the Ben Derrett's example shows. – Kolmo Mar 12 '12 at 10:50
If $A_1$, $A_2$, and $A_3$ are events (not random variables), then it is true that $A_1$, $A_2$, and $A_3$ are said to be pairwise independent events if the three conditions $$P(A_iA_j) = P(A_i)P(A_j), ~ i \neq j,$$ hold. This looks sort of similar to your mistaken assertion about random variables. But even in this case, it is not true that $A_1$, $A_2$, and $A_3$ are said to be mutually independent events if $$P(A_1A_2A_3) = P(A_1)P(A_2)P(A_3)$$ holds. All four displayed equations must hold for $A_1$, $A_2$, and $A_3$ to be mutually independent events. – Dilip Sarwate Mar 12 '12 at 11:25

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