The joint probability distribution of random variables is the probability distribution of the random vector . It records both the marginal distributions and their dependence.
A family of random variables is exchangeable if its joint probability distribution is unchanged by finite permutations of its coordinates. Independent and identically distributed random variables are exchangeable, but exchangeable random variables need not be independent random variables.
For an infinite sequence of real exchangeable random variables, conditional on its invariant sigma-algebra of an exchangeable sequence, the coordinates are independent and identically distributed. In particular, for Borel sets ,Here is the core proof. Let denote invariance under permutations of coordinates . Symmetrization as conditional expectation identifies the empirical average of each indicator with its conditional expectation on . The reverse martingale convergence theorem gives its limit conditional on . Multiplying the averages sums over all ordered coordinate tuples, including repetitions. Symmetrization of the product itself sums over the distinct tuples. The fraction of repeated tuples is at most , so these bounded averages have the same limit. This proves the displayed factorization. Permutation invariance also makes the individual conditional laws identical. A regular conditional distribution on the real line packages that common law as a random probability measure, giving the equivalent mixture-of-iid formulation.
On the coordinate product space of a sequence, the invariant sigma-algebra consists of measurable events unchanged by every finite permutation of coordinates. Its definition is pointwise and does not depend on the probability law. An exchangeable law makes each such permutation measure preserving. Events in the tail sigma-algebra are invariant because any finite permutation fixes every sufficiently late coordinate. Conversely, exchangeable invariant events are tail events modulo null sets.
For a real sequence of exchangeable random variables, every event in its invariant sigma-algebra of an exchangeable sequence agrees almost surely with an event of its tail sigma-algebra. Put . Approximate in by its conditional expectation on the first coordinates. Swap that block with the next coordinates. Invariance of and the law shows that the conditional expectation on the second block is an equally good approximation. It is measurable with respect to the tail after index . The L1 contraction of conditional expectation then bounds by twice the approximation error. The reverse martingale convergence theorem gives almost surely. Thresholding the latter at supplies the required tail event. Together with the reverse inclusion, the two sigma-algebras have identical probability completions.
For a taxi starting at any vertex of a rectangle with side lengths , a uniform pickup vertex gives approach-distance law uniform on . Conditional on that pickup, a uniform distinct destination gives passenger-distance law uniform on , independent of the pickup vertex. Hence the two distance variables are independent despite sharing that vertex. Their convolution has twelve equally likely pairs. Its mean is and its variance is . Repeated numerical sums combine their multiplicities when some distances coincide.
A marginal distribution is the distribution of a selected coordinate or subvector of a random vector, obtained from the joint distribution by integrating or summing over the remaining coordinates.
Marginalization sums or integrates a joint probability distribution over unwanted coordinates to obtain a marginal distribution.
A marginal density is a probability density function for one component of a joint probability distribution. Integrating the joint probability density over the other coordinates gives it; for two components, .
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