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Math Help - Number of records (expected value, variance)

  1. #1
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    Number of records (expected value, variance)

    Let X_1, X_2, ... , X_n be a sequence of independent identically distributes continous random variables. A record occurs at time j (  j \leq n) if X_j \geq X_i \; \forall \; 1 \leq i \leq j.
    Show that:
    \mathbb{E}[number of records]= \sum_{j=1}^{n} \frac{1}{j}
    \mathbb{D}^2[number of records]= \sum_{j=1}^{n} \frac{1-j}{j^2}

    Thank you very much in advance!
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  2. #2
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    Re: Number of records (expected value, variance)

    First af all, you should calculate:
    a) P(X_n is a record)=?
    then:
    b) E [number of records by time n]=?
    c) Var [number of records by time n]=?
    I give you some hints how to solve these problems:
    For part (a) use symmetry and observe that P(X_i = X_j ) = 0
    since X_iís are continuous random variables. For parts (b) and (c) introduce indicator random variables. Note that these indicator random variables are independent
    in this case.

    I hope I helped you.
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  3. #3
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    Re: Number of records (expected value, variance)

    Thank you for your help!
    On the other hand, to tell the truth, I still can't solve the problem.
    I would really appreciate any help!
    Last edited by zadir; November 27th 2011 at 09:20 AM. Reason: typo
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