## random variable with "average distribution"

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• 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
Message 1 of 2 , Jan 5, 2002
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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
• 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
Message 2 of 2 , Jan 6, 2002
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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
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