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Math Help - Probability theory question

  1. #1
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    Probability theory question

    Let  \theta (x) be a random variable and  \overline{\theta} (x) be the expected value. Moreover  S is defined:

     S = \{ x : \theta (x) < \frac{\overline{\theta} (x)}{p} \}

    Then the probability measure of  S is:

     \mu (S) \ge (1-p)

    Prove...

    It can be noted that there exist both density and cumulative distribution functions over  S .

    -----------------------------------------------------------------------------------

    I have been working on this problem a bit more and found some more info. It seems it is an application of the Markov Property, which states:

    Let  X be a measurable set with the characteristic function  \chi = 1 if  x \epsilon A and  \chi = 0 otherwise, where  A = \{ x \epsilon X : | f | \geq t  \} then:

     \mu (A) \leq \frac{1}{t} \int_x | f | d \mu

    Maybe it is obvious from this but I cannot work out the final details ....

    A proof of the markov inequality can be seen on wikipedia..
    Last edited by gloiterbox; June 30th 2011 at 11:58 PM.
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  2. #2
    Lord of certain Rings
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    Re: Probability theory question

    Quote Originally Posted by gloiterbox View Post
    Let  \theta (x) be a random variable and  \overline{\theta} (x) be the expected value. Moreover  S is defined:

     S = \{ x : \theta (x) < \frac{\overline{\theta} (x)}{p} \}

    Then the probability measure of  S is:

     \mu (S) \ge (1-p)

    Prove...

    It can be noted that there exist both density and cumulative distribution functions over  S .

    -----------------------------------------------------------------------------------

    I have been working on this problem a bit more and found some more info. It seems it is an application of the Markov Property, which states:

    Let  X be a measurable set with the characteristic function  \chi = 1 if  x \epsilon A and  \chi = 0 otherwise, where  A = \{ x \epsilon X : | f | \geq t  \} then:

     \mu (A) \leq \frac{1}{t} \int_x | f | d \mu

    Maybe it is obvious from this but I cannot work out the final details ....

    A proof of the markov inequality can be seen on wikipedia..
    Hint: Apply Markov inequality to S complement
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  3. #3
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    Joined
    Sep 2008
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    Re: Probability theory question

    Haha yes that works, nice trick Isomorphism!!!
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