I have answered my own question - but for posterity:

Thomas & Cover's "Elements of Information Theory" lays the

groundwork, p. 30; what follows is a generalization.

Consider a sequence of random variables X1...Xm (read: X sub 1

through X sub m) on a set A with distributions p1...pm. Let Q be a

random variable on the integers 1...m such that Pr(Q=j) = qj. Then

consider the random variable Z = X_{Q} (read: X sub Q).

The distribution of Z is \sum_{j} qj * pj, the average of the

distributions of the X's weighted by the distribution of Q. Z is the

random variable I was seeking, and it is indeed (somewhat) well-known.

- A

--- In probability@y..., "allan.drummond" <allan.drummond@t...> wrote:

> Let X1,...,Xm be a sequence of random variables on a probability

> space P with values drawn from an alphabet A.

>

> Let Pr(xij) be the probability that the j'th random variable takes

on

> the i'th value of A.

>

> Let Q = {(q1,...,qm) elem R^m : 0 <= qj <= 1; \sum qj = 1}.

>

> I'm looking for a random variable Z, also taking values from A,

such

> that Pr(zi) = \sum_j qj Pr(xij). In other words, I'm looking for a

> random variable Z whose distribution is equal to the mean

> distribution of the Xj under Q.

>

> Is there some well-known transformation of random variables that

does

> this? Can anyone point me to a resource that has more information

on

> this kind of thing?

>

> Thanks,

> Allan