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 Probabilities of probabilities

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Posts : 184
Join date : 2009-09-15
Age : 30
Location : the infinite, frictionless plane of uniform density

PostSubject: Probabilities of probabilities   Wed Sep 16, 2009 12:24 am

Suppose we have a set of two elements, say S={1,2}. Consider the set P(S) of all probability spaces having S as sample space. Then an element of P(S) is completely determined by the probability of the event {1}. In fact there is a bijection f : [0,1] -> P(S), sending t to the unique element of P(S) in which the event {1} has probability t. In other words P(S) is one-dimensional and looks like a line segment.

Now we can turn P(S) into a probability space itself, by taking the uniformly distributed random variable Z on the interval [0,1] and mapping it using the above bijection to P(S). Thus we can obtain a random variable Y, whose value is a probability space on the set S={1,2}. Moreover Y is "uniformly distributed" over P(S).

Now one can ask various questions : what is the expected value of the expected value of Y? What is the variance of the variance of Y? How can the above construction be generalized? What can we say about it?

(I propose the above problem more as a fun research problem than as a challenge.)
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