Probability Distribution with certain Mutual Information Condition

Question

I'm looking for an example of a probability distribution on X,Y,Z (any three r.v.'s) with $$I(X;Z) + I(Y;Z) < I(X,Y;Z)$$ where $I$ is standard mutual information. How might I go about this? Thanks!

Answer 1

Taking $X,Y,Z$ to be pairwise independent but not mutually independent should work

For example, each of the following cases with probability $\frac14$:

X   Y   Z  
---------
0   0   0

0   1   1

1   0   1

1   1   0