Results 1 to 13 of 13

Math Help - expectaion^2 and variance^2

  1. #1
    Member
    Joined
    Mar 2008
    From
    Texas
    Posts
    110
    Awards
    1

    expectaion^2 and variance^2

    I need help solving these:

    E\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2 \right]

    Var\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2\right]

    (btw, what the code to make both delimiters tall.)

    :-)

    Thanks.
    Last edited by CaptainBlack; December 12th 2009 at 02:00 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by yvonnehr View Post
    I need help solving these:

    E\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2 \right]

    Var\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2\right]

    (btw, what the code to make both delimiters tall.)

    :-)

    Thanks.
    Expand the expression inside the expectation and then use known properties of expectation to simplify (mainly linearity)

    For the variance re write as an expectation and repeat the above process.

    \left( \right)

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    BY any chance, are you sampling from a normal population with n=36?
    If so, the rv inside the square root is a st normal, which makes your life easier.
    Even if n isn't 36, normality makes this a lot easier.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Mar 2008
    From
    Texas
    Posts
    110
    Awards
    1

    Talking n=36, Thanks!

    Yes! n=36.

    Thanks guys! I'll work on this and see how it goes.

    :-)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    If the 36 observations are iid Normals, then

    \frac{\overline{X}-\mu}{\sigma/6}\sim N(0,1)

    and you just need the distribution of a st normal squared, which is a Chi-Square with 1 df.
    And the mean and variance of a chi-square is trivial.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Mar 2008
    From
    Texas
    Posts
    110
    Awards
    1

    Question Z^2 ~ X^2 df 1 - Why?

    Quote Originally Posted by matheagle View Post
    If the 36 observations are iid Normals, then

    \frac{\overline{X}-\mu}{\sigma/6}\sim N(0,1)

    and you just need the distribution of a st normal squared, which is a Chi-Square with 1 df.
    And the mean and variance of a chi-square is trivial.
    Matheagle,
    So I finally found a reference that tells me that the distribution of Z^2 is Chi-squared with df 1. But I didn't find anything that shows me how or why that works.

    Will you tell me or point me to somewhere I can see a proof or some explanation? (You must pardon me but I am a math major and like to see how things work.)

    Thanks a bunch for your help!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    All that work is a waste of time if X_1,...,X_{36} are NOT from a normal population.
    I asked if they were and I don't see a response.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Mar 2008
    From
    Texas
    Posts
    110
    Awards
    1

    Response to ? about normal pop.

    Quote Originally Posted by matheagle View Post
    All that work is a waste of time if X_1,...,X_{36} are NOT from a normal population.
    I asked if they were and I don't see a response.
    Yes, it is from a normal population.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    all this is in many books, including walpole.
    Use MGFs to prove that the linear combination of normals is normal
    and then use that 2-1 transformation to prove that the square of a st normal is a chi-square.
    I bet I can find it online.
    I can prove it without looking it up, but I want to make dinner and grade exams tonight.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member
    Joined
    Mar 2008
    From
    Texas
    Posts
    110
    Awards
    1

    Thanks for the proof outline.

    Quote Originally Posted by matheagle View Post
    all this is in many books, including walpole.
    Use MGFs to prove that the linear combination of normals is normal
    and then use that 2-1 transformation to prove that the square of a st normal is a chi-square.
    I bet I can find it online.
    I can prove it without looking it up, but I want to make dinner and grade exams tonight.
    Okay, great. I can try it/find it with the outline you have given. Thanks!

    Have a good dinner.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    I couldn't find it, but I'll start it for you.

    Let z be a st normal and X=Z^2

    F_X(x)=P(X\le x)=P(Z^2\le x)=P(-\sqrt{x}\le Z\le \sqrt{x} )=F_Z(\sqrt{x})-F_Z(-\sqrt{x})

    NOW differentiate wrt x and obtain the realtionship between the two densities and then plug in.
    SHOW me your work.

    {d\over dx}\Biggl(F_X(x)=F_Z(\sqrt{x})-F_Z(-\sqrt{x})\Biggr)

    f_X(x)=f_Z(\sqrt{x}) {1\over 2\sqrt{x}} -f_Z(-\sqrt{x}){-1\over 2\sqrt{x}}

    =\biggl(f_Z(\sqrt{x})+f_Z(-\sqrt{x})\biggr){1\over 2\sqrt{x}}

    NOW plug in.
    Last edited by matheagle; December 12th 2009 at 07:35 PM.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Member
    Joined
    Mar 2008
    From
    Texas
    Posts
    110
    Awards
    1

    Ha ha!

    Quote Originally Posted by matheagle View Post
    I couldn't find it, but I'll start it for you.
    Let z be a st normal and X=Z^2
    Oh, you are real funny! That's a good one!!! The wit of a teacher is unmistakable.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by yvonnehr View Post
    Matheagle,
    So I finally found a reference that tells me that the distribution of Z^2 is Chi-squared with df 1. But I didn't find anything that shows me how or why that works.

    Will you tell me or point me to somewhere I can see a proof or some explanation? (You must pardon me but I am a math major and like to see how things work.)

    Thanks a bunch for your help!
    See post #7 in this thread: http://www.mathhelpforum.com/math-he...tml#post119542
    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, 02:53 AM
  2. variance of the sample variance
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: February 8th 2011, 09:35 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, 06:39 AM
  4. Variance Help
    Posted in the Statistics Forum
    Replies: 2
    Last Post: August 23rd 2009, 02:22 PM
  5. Variance of the population variance.
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: November 10th 2008, 10:07 AM

Search Tags


/mathhelpforum @mathhelpforum