Page 73 - Elements of Distribution Theory

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2.5 Exchangeability 59

Note that E(Y|X) = µ, where µ = E(Y). Then, according to Corollary 2.2, for any real-
valued function h of Y,
2
2
E[(h(X) − Y) ] ≤ E[(Y − µ) ];
that is, the best approximation to Y among all functions of X is simply the constant function
h(X) = µ.

4
Example 2.18. Let X and Y denote real-valued random variables such that E(X ) < ∞
4
and E(Y ) < ∞. Suppose that Y = X + Z where Z is a real-valued random variable with
2
E(Z) = 0 and E(Z ) = 1; assume that X and Z are independent.
Then, according to Corollary 2.2, the best approximation to Y among functions of X is
E(Y|X) = E(X + Z|X) = X.
2
The best approximation to Y among functions of X is
2
2
2
E(Y|X) = E[(X + Z) |X) = E(X |X) + E(2XZ|X) + E(Z |X)
2
2
= X + 2XE(Z|X) + E(Z )
2
= X + 1.
2
Hence, although the best approximation to Y is X, the best approximation to Y is
2
2
X + 1, not X . This is due to the criterion used to evaluate approximations.
2.5 Exchangeability
Recall that random variables X 1 , X 2 ,..., X n are independent and identically distributed if
they are independent and each X j has the same marginal distribution. An infinite sequence
of random variables X 1 , X 2 ,... is independent and identically distributed if each finite
subset is.
Exchangeability provides a useful generalization of this concept. Recall that a permuta-
tion of (1, 2,..., n)isa rearrangement of the form (i 1 ,..., i n ) such that each 1 ≤ i j ≤ n
is an integer and i j = i k for j = k. Real-valued random variables X 1 ,..., X n are said
to have an exchangeable distribution or, more simply, to be exchangeable if the distribu-
) for any permutation
tion of (X 1 ,..., X n )is the same as the distribution of (X i 1 ,..., X i n
(i 1 , i 2 ,..., i n )of(1, 2,..., n).
As noted above, the simplest example of exchangeability is the case of independent,
identically distributed random variables. A formal statement of this is given in the following
theorem; the proof is straightforward and is left as an exercise.

Theorem 2.7. Suppose X 1 ,..., X n are independent identically distributed random vari-
ables. Then X 1 ,..., X n are exchangeable.

Example 2.19 (Bivariate distribution). Consider the distribution considered in Exam-
ples 2.1 and 2.11. The random vector (X, Y) has an absolutely continuous distribution
with density function

p(x, y) = 6(1 − x − y), x > 0, y > 0, x + y < 1.