Results 1 to 3 of 3

Math Help - Variance

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Variance

    \displaystyle f(x)=\frac{1}{2^{n/2}\Gamma\left(\frac{n}{2}\right)}x^{\frac{n}{2}-1}e^{-x/2}

    \displaystyle M(t)=\text{E}[e^{tX}]=\frac{1}{2^{n/2}\Gamma\left(\frac{n}{2}\right)}\lim_{b\to\infty}  \int_0^bx^{\frac{n}{2}-1}e^{-x\left(\frac{1}{2}-t\right)} \ dx

    \displaystyle =\frac{1}{2^{n/2}\Gamma\left(\frac{n}{2}\right)}\Gamma\left(\frac  {n}{2}\right)\left(\frac{1}{\frac{1}{2}-t}\right)^{\frac{n}{2}}=(1-2t)^{-\frac{n}{2}}

    \displaystyle\left[\frac{dM(t)}{dt}\right]_{t=0}=n

    \displaystyle\left[\frac{d^2M(t)}{dt^2}\right]_{t=0}=n(n+2)

    \text{Var}[X]=n^2+2n-n^2=2n

    Correct?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Oct 2009
    Posts
    340
    I wouldn't have done it that way, but yes, it is correct.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    By the way this is a \Gamma({n\over 2},2)=\chi^2_n random variable.

    You can prove that the MGF of a \Gamma(\alpha, \beta) random variable is (1-\beta t)^{-\alpha}
    Then plug in the n/2 and 2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with variance
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: September 3rd 2011, 01:53 AM
  2. variance of the sample variance
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: February 8th 2011, 08:35 PM
  3. How to get new variance given old variance and average
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 23rd 2010, 05:39 AM
  4. Variance Help
    Posted in the Statistics Forum
    Replies: 2
    Last Post: August 23rd 2009, 01:22 PM
  5. Variance of the population variance.
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: November 10th 2008, 09:07 AM

/mathhelpforum @mathhelpforum