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Combined sample spaces .

  1. Feb 20, 2008 #1
    Dear all

    I've just begun studying measure theory , and i cant help it but to think of it in terms of probability theory , i don't know if that is right or wring . any way , i have this naive question :

    consider the following : we have n sample spaces [tex]\Omega_{}i[/tex], each with a distribution P[tex]_{}i[/tex] ( i=1,...n) , if we combine (union) the sample spaces to form a new sample space whose distribution is unknown , is there a way to extract the distribution of the new sample space from the previously know distributions ??
     
  2. jcsd
  3. Feb 20, 2008 #2

    EnumaElish

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    The simplest case is to assume disjoint sample spaces. In that case, the probability of obtaining element x from the k'th space, x(k), will be P{x(k)} = P{x|k}P{k} = Pk{x}P{k}, where P{k} is the probability of obtaining the k'th space within the set of all spaces, or the measure of the k'th space in the union.

    In general:

    [tex]P\{x\} = \sum_{k=1}^N P\{x|k\}P\{k\}[/tex]

    where N is the number of spaces.

    This is related to the axiom of independence [from] irrelevant alternatives in decision theory.
     
    Last edited: Feb 21, 2008
  4. Feb 20, 2008 #3
    what about the general case ?
     
  5. Feb 20, 2008 #4

    EnumaElish

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    See the portion of my post that begins with "In general."
     
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