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Modelling probability spaces

  1. Apr 14, 2010 #1
    measure spaces and random variable are still fuzzy to me although i'm starting to see how random variable are s
    used, i need some clarification on the following

    suppose we have a random variable [tex] X [/tex] we don't know what the exact sample space is but say we know the density is [tex] f_{X}(x) , x \in [a,b] [/tex]
    now can we re-model the probability space as
    [tex] \Omega = [a,b] [/tex]
    sigma field [tex] \mathcal{F} = \mathcal{B}([a,b]) [/tex] and probability measure [tex] P(M) = \int_{M}f_{X}(x) dm [/tex]

    Can i assume that the borel set generated by [a,b] is a valid sigma field, since [tex] X [/tex] has a density defined on [a,b], so the measure must be defined on all [tex] \mathcal{B}([a,b]) [/tex] as well (not sure how to prove this formally, if this is correct)
    suppose now if we had a Random variable [tex] Y = 2X [/tex]
    then we could say the distribution of Y [tex] P_{Y}(B) = P(\{ Y \in B \}) = \int_{K}f_{X}(x) dm [/tex] where [tex] K = \{ k : 2k \in B \} [/tex].
  2. jcsd
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