# Probability distribution functions: factorization 3-way implies 2-way?

I recently asked a question about pairwise versus mutual independence (also related to this and this q).

However,

(1) I inadvertently used incorrect terminology:

three events, A, B, C are mutually independent when:

P[A,B]=P[A]P[B], P[B,C]=P[B]P[C], P[A,C]=P[A]P[C], P[A,B,C]=P[A]P[B]P[C]

Did and others pointed out that

"Mutual independence means the four identities you copied, pairwise independence means the first three of these identities." -- Did

Note that the term mutual has varying definitions across math. For example, mutual information is a pairwise relation.

(2) Going back to probability, GC Rota said the theory can be approached two ways: focusing on random variables (event algebra) or focusing on distributions. Here I am interested in distributions, where independence can be interpreted as factorization of the probability distribution function. The conditions are the same as above, where P is interpreted as the PDF function.

The following graphic based on a standard example from Counterexamples in Probability and Statistics of a 3-dimensional binomial PDF that factorizes pairwise (ie, each of the 3 pairs of random variables are independent and the 2-dim joint distributions can all be written as the product of the respective marginals) but not 3-way independent (the joint distribution cannot be written as the product of the individual marginal distributions)

My question is whether the opposite can happen, ie if the 3-dim (or perhaps higher) joint distribution factorizes into the 1-dim marginals, does that imply the pairwise factorization of all 2-dim joint distributions into the marginals?

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Could you perhaps phrase the question in terms of PDFs? At the moment this sounds a lot like asking: if three variables $X_1,X_2,X_3$ are mutually independent, then are $X_1$ and $X_2$ independent? –  Colin McQuillan Feb 3 '13 at 1:14
@ColinMcQuillan, It's not just about the independence of $X_1$ and $X_2$ but all pairs formed from the set $\{X_1,X_2,X_3\}$. Or do you have an issue with the R.V. versus distributional definition? Also, as I state above, mutual statistical independence includes independence of the pairs not just the joint, so I am avoiding that definition to focus on the 3-way to 2-way implication, which is unambiguous. –  alancalvitti Feb 3 '13 at 2:23
Got something from an answer below? –  Did Feb 9 '13 at 10:25
Yes, thank you Did - both your and Dilip's answers are good, and although his displays the marginalization directly, I accepted yours because it is more general of the two. –  alancalvitti Feb 9 '13 at 21:00

Indeed, assume that $\mu$ is the product of the probability measures $\mu_1$, $\mu_2$ and $\mu_3$, hence, for every $(A_1,A_2,A_3)$, $\mu(A_1\times A_2\times A_3)=\mu_1(A_1)\cdot\mu_2(A_2)\cdot\mu_3(A_3)$.
Then, for example, the 2-marginal distribution $\nu$ corresponding to the two first coordinates is such that $\nu(A)=\mu(A\times\mathbb R)$ for every 2-dimensional $A$.
One sees that, for every $(A_1,A_2)$, $\nu(A_1\times A_2)=\mu(A_1\times A_2\times\mathbb R)=\mu_1(A_1)\cdot\mu_2(A_2)$, that is, that $\nu$ is indeed the product of $\mu_1$ and $\mu_2$.
If the joint density of $X$,$Y$, and $Z$ is $f_{X,Y,Z}(x,y,z)$, then for any pair, say $Y$ and $Z$, $$f_{Y,Z}(y,z) = \int_{-\infty}^{\infty}f_{X,Y,Z}(x,y,z)\,\mathrm dx.$$ Now suppose that $f_{X,Y,Z}(x,y,z) = f_X(x)f_Y(y)f_Z(z)$ for all real numbers $x,y$ and $z$. Substitution in the integral gives $$f_{Y,Z}(y,z) = \int_{-\infty}^{\infty}f_X(x)f_Y(y)f_Z(z) \mathrm dx = f_Y(y)f_Z(z)\int_{-\infty}^{\infty}f_X(x)\mathrm dx = f_Y(y)f_Z(z).$$ I will leave it to you to verify that a similar result holds for the other two pairs of random variables.