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Homework Help: Question about probability measure

  1. Sep 7, 2005 #1
    An probability measure on same space [itex]\Omega[/itex] is a function of subsets of [itex]\Omega[/itex] satisfying three axioms:

    (i) For every set [itex]A \subset \Omega[/itex], the value of the function is a non-negative number: P(A) [itex]\geqslant[/itex] 0.

    (ii) For any two disjoint sets A and B, the value of the function for their union A + B is equal to the sum of its value for A and its value for B:

    P(A + B) = P(A) + P(B) provided A.B = [itex]{\O}[/itex].

    (iii) The value of the function for [itex]\Omega[/itex] (as a subset) is equal to 1:

    P([itex]\Omega[/itex]) = 1.


    Now, reply these questions:

    If M is a probability measure, show:

    (a) that the function M/2 satisfies Axiom(i) and (ii) but not (iii).

    (b) the function [itex]M^2[/itex] satisfies (i) and (iii) but not necessary (ii); give a counterexample to (ii).
     
  2. jcsd
  3. Sep 7, 2005 #2

    Hurkyl

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    We can't help if you if you haven't tried it.
     
  4. Sep 7, 2005 #3

    David

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  5. Sep 8, 2005 #4
  6. Sep 8, 2005 #5

    sic

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    Omega={A,B}
    M(gurnisht)=0
    M({A})=1/2
    M({B})=1/2
    M({A,B})=1
    then
    M^2({A,B})!=M^2({A})+M^2({B})
    (Either it's that simple or i don't understand the problem)
     
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