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Math Help - finding expected value

  1. #1
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    finding expected value

    Let X_1,....,X_n be iid rv with density f and cumulative distribution function F. Let:

    I_{X1}(a)=1, (\text{if } X_1\leq a) \text{ and}=0, \text{ (otherwise)}

    I want to find the expected value of I_{X1}(2)

    I can't seems to find the answer since the distribution is unknown to me
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by noob mathematician View Post
    Let X_1,....,X_n be iid rv with density f and cumulative distribution function F. Let:

    I_{X1}(a)=1, (\text{if } X_1\leq a) \text{ and}=0, \text{ (otherwise)}

    I want to find the expected value of I_{X1}(2)

    I can't seems to find the answer since the distribution is unknown to me
    The expected value of an indicator function is just the probability of that event.

    Hence E(I_{X1}(2))=P(X1\le 2)=F_{X1}(2)
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  3. #3
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    Suppose now p=P(X1\leq 2). How to find the mle (maximum likelihood estimate) of p, based on the whole sample.
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  4. #4
    MHF Contributor matheagle's Avatar
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    I would think that these are i.i.d. Bernoulli's.

    Let Y_i=I(X_i\le 2) for i=1,2,...,n.

    Here p=P(X1\le 2) isn't really the point

    The MLE of Bernoullis is \hat p the sample mean of the Y's.

    \hat p={\sum_{i=1}^nI(X_i\le 2)\over n}

    Since the Likelihood function is

    L= p^{Y_1}(1-p)^{1-Y_1}p^{Y_2}(1-p)^{1-Y_2}\cdots p^{Y_n}(1-p)^{1-Y_n}

    = p^{\sum_{i=1}^n Y_i}(1-p)^{n-\sum_{i=1}^n Y_i}

    Now take the log and differentiate TWICE and obtain the sample mean as the MLE.
    Last edited by matheagle; November 21st 2009 at 09:28 PM.
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  5. #5
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    Given the Y_i defined by u:

    Let say now I want to know the MLE of p(1-p), is it right for me to conclude that it is \overline{Y}(1-\overline{Y})?
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  6. #6
    MHF Contributor matheagle's Avatar
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    I believe that is correct under the invariance principle of MLEs.
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