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Math Help - Mean and Variance problem

  1. #1
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    Mean and Variance problem

    Let T=  \sum ln(1+x_i)

    Find the mean and variance of T.

    some other info: X_1,X_2....X_n are iid rv with density function  f(x,h)= h(1+x) ^\ (-h-1) , x>=0 and h>0, h is an unknown variable

    I'm not really sure how to approach this problem
    Any help will be appreciated

    Thanks in advance
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  2. #2
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    Obtain the mean and variance of any particular ln(1+x):

    Define Y_i=ln(1+x_i)
    E(Y_i) = \int f(x_i,h)ln(1+x) dx_i

    E^2(Y_i) = \int f(x,h)\left( ln(1+x)\right) ^2 dx

    Var(Y_i) = E^2(Y_i) - \left( E(Y_i) \right)^2

    I haven't checked but those integrals look do-able to me.


    Then,Use the standard results for summing independant variables
    E \left( \sum \left(Y_i \right) \right)= \sum E \left(Y_i \right)...

    Var \left( \sum \left(Y_i \right) \right)= \sum Var \left(Y_i \right)
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  3. #3
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    I got

     E(Y_i) = {-(x+1)^ {-h} (h(ln(x+1)+1) * \frac {1}{h}
     E(Y_i)^2 = {-(x+1) ^ {-h} (h^2 (ln^2(x+1) +2h ln (x+1)+2)) * \frac {1}{h^2}
    from wolfram

    For  \sum E(Y_i) , it is just  {-\sum(x+1)^ {-h} (h \sum(ln(x+1) +1) * \frac {1}{h}
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  4. #4
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    are you supposed to use wolfram to do your integrals?

    anyway, for the last step you are expected to notice that all the Y_i are iid, so

    \displaystyle \sum_{i=1}^{n} E(Y_i) = nE(Y)

    \displaystyle \sum_{i=1}^{n} Var(Y_i) = nVar(Y)
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