Page 74 - Elements of Distribution Theory

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052184472Xc02 CUNY148/Severini May 24, 2005 2:29

60 Conditional Distributions and Expectation

2
Let g denote a bounded, real-valued function defined on (0, 1) . Then
1 1
E[g(X, Y)] = g(x, y)6(1 − x − y)I {x+y<1} dx dy
0 0
1 1
= g(y, x)6(1 − y − x)I {y+x<1} dy dx = E[g(Y, X)].
0 0
It follows that (Y, X) has the same distribution as (X, Y)so that X, Y are exchan-
geable.

Example 2.20. Let (X 1 , X 2 ) denote a two-dimensional random vector with an absolutely
continuous distribution with density function

1
p(x, y) = , 0 < x 2 < x 1 < 1.
x 1
Note that X 2 < X 1 with probability 1. Let (Y 1 , Y 2 ) = (X 2 , X 1 ). Then

Pr(Y 2 < Y 1 ) = 0.
It follows that (Y 1 , Y 2 ) does not have the same distribution as (X 1 , X 2 ); that is, (X 2 , X 1 )
does not have the same distribution as (X 1 , X 2 ). It follows that the distribution of (X 1 , X 2 )
is not exchangeable.

Suppose that X 1 , X 2 ,..., X n are exchangeable random variables. Then the distribution
of (X 1 , X 2 ,..., X n )is the same as the distribution of (X 2 , X 1 ,..., X n ). In this case,

Pr(X 1 ≤ x, X 2 < ∞,..., X n < ∞) = Pr(X 2 ≤ x, X 1 < ∞,..., X n < ∞).
That is, the marginal distribution of X 1 is the same as the marginal distribution of X 2 ;it
follows that each X j has the same marginal distribution. This result may be generalized as
follows.

Theorem 2.8. Suppose X 1 ,..., X n are exchangeable real-valued random variables. Let m
denote a positive integer less than or equal to n and let t 1 ,..., t m denote distinct elements
) does not depend on the choice of
of {1, 2,..., n}. Then the distribution of (X t 1 ,..., X t m
t 1 , t 2 ,..., t m .
Proof. Fix m and let t 1 ,..., t m and r 1 ,...,r m denote two sets of distinct elements from
{1, 2,..., n}. Then we may find t m+1 ,..., t n in {1,..., n} such that (t 1 ,..., t n )isa permu-
tation of (1, 2,..., n); similarly, suppose that (r 1 ,...,r n )isa permutation of (1, 2,..., n).
)have the same distribution. Hence,
Then (X t 1 ,..., X t n ) and (X r 1 ,..., X r n
< ∞)
Pr(X t 1 ≤ x 1 ,..., X t m ≤ x m , X t m+1 < ∞,..., X t n
< ∞);
= Pr(X r 1 ≤ x 1 ,..., X r m ≤ x m , X r m+1 < ∞,..., X r n
the result follows.

Thus, exchangeable random variables X 1 ,..., X n are identically distributed and any two
subsets of X 1 ,..., X n of the same size are also identically distributed. However, exchange-
able random variables are generally not independent.