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Math Help - Expected Value and variance

  1. #1
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    Expected Value and variance

    Independent observations X1......Xn are such that Xi belongs to {0.,1} and for some 0<theta<1, it is that Pr(Xi=1)=theta.

    Let n1 be the number equal to 1.

    Find the expected value and variance of n1?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Expected Value and variance

    Let's use k instead of n_{1} and p instead of \theta because they are 'more familiar'. The probability to have k ones among n variables is...

    P_{n,k}= a_{n,k}\ p^{k}\ (1-p)^{n-k} (1)

    ... where...

    a_{n,k} = \frac{n!}{k!\ (n-k)!} (2)

    The expected value of k is then...

    \mu= \sum_{k=0}^{n} k\ P_{n,k} (3)

    ... and the variance...

    \sigma= \sqrt{\sum_{k=0}^{n} (k-\mu}^{2}\ P_{n,k}} (4)



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    Re: Expected Value and variance

    Thank you so much
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