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Meaning of probability measure

  1. Mar 11, 2009 #1
    What does it mean by
    [tex]\int f(w) P (dw)[/tex]

    I don't really understand [tex] P (dw) [/tex] here. Does it mean [tex] P (x: x \in B(x, \delta))[/tex] for infinitely small [tex] \delta [/tex]?

    For example, with [tex] P(x)=1/10[/tex] for [tex] x=1, 2, ..., 10 [/tex]. How can we interpret this in term of the above integral

    Thanks...
     
  2. jcsd
  3. Mar 11, 2009 #2
    I never took measure theory. Therefor my totally naive answer would be P(dw) is just a distribution function. Of course I'm probably completly wrong given I don't even know what a measure is.
     
  4. Mar 11, 2009 #3
    P(dw) is like a distribution fuction... may be. I am confused about P(dw), is it probability of dw? Then what is dw? Following the above example, say, we have [tex] f(w)=1 [/tex] for w=1 and 0 otherwise. What is the meaning of dw here and hence value of P(dw) at w=1? I guess the above integral would give value = 1/10.
     
  5. Mar 11, 2009 #4

    HallsofIvy

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    I have never seen "P(dw)". I think you mean what I would call dP(w)= P'(w)dw- the derivative of the cumulative probability distribution and so the probability density function. In that case, [itex]\int F(w)dP= \int F(w)P'(w)dw[/itex] is the expected value of F.
     
  6. Mar 11, 2009 #5
    I guess if P(w) has a derivative we can write it that way. I got the expression from a text book by Patrick Billingsley. Generally, when P(w) is not differentiable (as shown in the example), we can not write the expression in that form.
     
  7. Mar 11, 2009 #6

    statdad

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    The notation
    [tex]
    \int_\Omega X(\omega}) \, \mathcal{P}(dw)
    [/tex]

    is used in probability to indicate the expectation of the random variable [tex] X [/tex]
    with respect tot the probability measure (distribution) [tex] \mathcal{P} [/tex] over the probability space [tex] \Omega [/tex].

    If [tex] \Lambda [/tex] is any measurable set, then

    [tex]
    \int_\Lambda X(\omega) \, \mathcal{P}(dw) = E[X \cdot 1_{\Lambda}]
    [/tex]

    If the probability space is the real line with measure [tex] \mu [/tex], then

    [tex]
    \int_\Lambda X(\omega) \, \mathcal{P}(dw) = \int_\Lambda f(x) \, \mu(dx)
    [/tex]

    is the Lebesgue-Stieltjes integral of [tex] f [/tex] with respect to the
    probability measure [tex] \mu [/tex].

    In more traditional form, if [tex] F [/tex] is the distribution function of [tex] \mu [/tex], and [tex] \Lambda [/tex] is an interval [tex] (a,b) [/tex], then

    [tex]
    \int_\Lambda X(\omega) \, \mathcal{P}(dw) = \int_\Lambda f(x) \, \mu(dx) = \int_{(a,b)} f(x) \, dF(x)
    [/tex]

    If the probability measure doesn't have any atoms, the final integral is just a Lebesgue integral. If there are atoms, you need to take care to specify the interval according to whether the endpoints are or are not included - e.g.

    [tex]
    \int_{a+0}^{b+0} f(x) \,dF(x), \quad \int_{a-0}^{b-0} f(x) \, dF(x)
    [/tex]

    and so on.

    Billingsley is one of the "classic" probability texts. Chang's "A Course in Probability Theory" is another - I studied from it many years ago, and have the second edition. His writing is a little terse, but there is a lot packed into his book.
     
  8. Mar 11, 2009 #7
    Thanks a lot for the detail answer. I guess by Chang you mean Kai Lai Chung... :)
     
  9. Mar 11, 2009 #8

    statdad

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    Yes, I did mean Kai Lai Chung - I would give a general description of my typing ability, but the description wouldn't be "safe for work".
    Sorry for the confusion - glad the answer helped.
     
  10. Mar 12, 2009 #9
    This isn't really correct. P may not be differentiable. When you say [itex]\int_B f(x) P(dx)[/itex], you are referring to the Lebesgue integral of f with respect to P. It is the same as saying [itex]\int_B f dP[/itex]. You are just telling where the arguments lie so there is no confusion. I have to disagree with statdad in that it is not the Stieltjes integral, it is just the plain old Lebesgue integral. For Stieltjes you want to take a distribution function F of P and then you work it out as [itex]\int_B f(x) dF(x)=\int_B f(x) P(dx)[/itex].

    Billingsley is a nice textbook and also I would recommend Ash, Real Analysis and Probability.
     
  11. Mar 12, 2009 #10
    Thanks...

    I will have a look at "Real Analysis and Probability" by Ash.
     
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