The joint distribution of random variables 
assigns probabilities
to events involving the variables simultaneously. For two
real-valued random
variables, it is determined by the joint
distribution function
 |
In the discrete case, the joint probability mass function is 
.
In the continuous case, when a joint probability
density function 
exists, probabilities are integrals
of 
over Borel subsets of the Euclidean
plane. Summing or integrating
out variables gives the marginal distributions.
A finite compatibility question can be expressed geometrically. Given real numbers 
,
,
,
and 
,
there are jointly distributed random variables 
, 
, and 
taking values in 
with 
, 
, 
, and 
iff
 |
Here the brackets denote expectation values. The eight deterministic vectors 
are the four mutually orthogonal vectors 
, 
, 
, and 
and the four vectors
obtained by multiplying them by 
. Their convex hull is a four-dimensional
cross polytope. Expressing 
in the orthogonal
basis formed by the four listed vectors gives the
displayed condition. In particular, no such distribution has 
and 
. This is a probability formulation of the compatibility
constraints used in the Greenberger-Horne-Zeilinger
theorem (de Barros and Suppes 2000, Trott 2006).
