Results 1 to 8 of 8

Math Help - Probability Density Function - Unbiased Estimator

  1. #1
    Junior Member
    Joined
    Dec 2008
    Posts
    70

    Probability Density Function - Unbiased Estimator

    Hi, I don't know how to solve this past exam question, and I can't find any course material that helps whatsoever

    Suppose that a random variable X has the probability density function
    f_{X}(x)=\left\{\begin{array}{cc}\frac{e^{{-x/\theta}}}{\theta},& \mbox{} x>0; {} \theta >0;}\\0,& \mbox{} otherwise.\end{array}\right.

    Consider a random sample X_{1}, X_{2},...,X_{n} from this distribution and let T=k\sum{X_{i}^2}

    Find k so that T is an unbiased estimator of \theta^2

    Thanks.

    p.s. the sum is i=1 to n, i don't know how to latex that
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    E(T) = E(k \sum(X_i^2)) = k \left( E(X_1^2) + E(X_2^2) + E(X_3^2) +.... +E(X_n^2) \right)

    But Xi are iid so:
    E(T) = nkE(X^2)

    E(T) = nk \int^\infty_0 x^2 f(x) dx

    integrate then choose the value of k such that E(T) = \theta^2
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2008
    Posts
    70
    thanks, although just one question, why do you multiply the integral part by x^2? I thought the estimator was found by just x, and E(X^2) was found by x^2?


    edit- don't worry, just realised i am being blind and it is infact a E(X^2)

    Thanks a lot!
    Last edited by Rapid_W; July 27th 2010 at 11:11 AM. Reason: oops
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Dec 2008
    Posts
    70
    i don't suppose anyone could clarify my answer of k=1/n could they?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    i should also mention that the pdf you ahve been given is for the exponential distribution.

    Depending on the preferences of your professor, you may be allowed to simply look up the fact that E(X^2) = 2\theta^2
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Dec 2008
    Posts
    70
    I ended up with E(T)=nk(\theta ^2) meaning i got E(X^2) = \theta ^2...damn,

    After integrating i got nk[-x^2e^{-x/\theta}-(\theta xe^{-x/\theta}-(-\theta ^2e^{-x/\theta}))]_{0}^\infty

    so either I can't substitute properly or I integrated wrong? Also, we are only issued the New Cambridge Statistical Tables by Lindley and Scott so we aren't supposed to rely on remembering the distributions.

    edit- OMG I think my brain has partially melted, I've been differentiating x^2 to just x, i feel so stupid right now :P
    Last edited by Rapid_W; July 27th 2010 at 12:32 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Dec 2008
    Posts
    70
    So if E(T)=nk(2\theta ^2) does this mean k=\frac {1}{2n} and that is the answer to the question?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    yep
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Is this an unbiased estimator for Xi?
    Posted in the Advanced Statistics Forum
    Replies: 12
    Last Post: October 14th 2011, 09:31 PM
  2. Unbiased Estimate of a probability density function
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 25th 2011, 02:40 PM
  3. unbiased estimator of μ1 μ2
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: March 24th 2010, 01:45 AM
  4. Unbiased Estimator
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: April 12th 2009, 08:53 AM
  5. unbiased estimator
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 20th 2008, 04:20 AM

/mathhelpforum @mathhelpforum