# pr.probability – Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail

Let $$X_1,ldots,X_n,Y,Z$$ be $$n+2$$ binary random variables and define $$X=(X_1,ldots,X_n)$$. In most problems, instead of treating $$X$$ as $$n$$ distinct binary random variables, there is no loss of generality in treating $$X$$ as a single variable $$U$$ that takes on $$2^n$$ states with the same probabilities (see below for a more rigourous interpretation). For example, $$Yperp Z|X iff Yperp Z|U$$, and quantities such as entropy and mutual information remain unchanged.

My question: Are there any examples where this replacement “fails”? That is, some property that holds for $$(X,Y,Z)$$ but doesn’t hold mutatis mutandis for $$(U,Y,Z)$$?

What I mean by “treating $$X$$ as a single variable $$U$$“:
More formally, let $$sigma:{1,ldots, 2^n}to {0,1}^n$$ be a bijection and enumerate the $$2^n$$ possible states of $$X$$ by $$sigma$$. We can define $$U$$ to be a random variable on $$2^n$$ states such that
$$P(U=k) = P(X=sigma(k)).$$