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Math Help - A couple questions with chi square

  1. #1
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    A couple questions with chi square

    1) Use the fact that (n-1)^2 S^2/ \sigma^2 is a chi square random variable with n-1 degrees of freedom to prove that
    Var(S^2)= 2\sigma^4/n-1.

    Hint: Use the fact that the variance of a chi square random variable with k degrees of freedom is 2k.

    2) Let V and U be independent chi square random variables with 7 and 9 degrees of freedom, respectively. Is it more likely that (V/7)/(U/9) will be between
    a) 2.51 and 3.29 or
    b) 3.29 and 4.20
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  2. #2
    MHF Contributor matheagle's Avatar
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    The square on (n-1) in your statement is incorrrect....

    Quote Originally Posted by Janu42 View Post
    1) Use the fact that {(n-1)S^2\over \sigma^2} is a chi square random variable with n-1 degrees of freedom to prove that
    Var(S^2)= 2\sigma^4/n-1.

    Hint: Use the fact that the variance of a chi square random variable with k degrees of freedom is 2k.

    2) Let V and U be independent chi square random variables with 7 and 9 degrees of freedom, respectively. Is it more likely that (V/7)/(U/9) will be between
    a) 2.51 and 3.29 or
    b) 3.29 and 4.20

     V(S^2) =  V\left (S^2{ (n-1)\sigma^2\over (n-1)\sigma^2}\right)

     = \left({\sigma^4\over (n-1)^2}\right) V\left (S^2{ (n-1)\over \sigma^2}\right)

     = \left({\sigma^4\over (n-1)^2}\right) V\left (\chi^2_{n-1}\right)

     = \left({\sigma^4\over (n-1)^2}\right) 2(n-1)

    (b) Use an F distribution table
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  3. #3
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    You're right, sorry i typoed. Thanks.

    For #2, do I use the pdf for a chi square?

    EDIT: Sorry, typed that while you were editing yours I guess. But OK, thanks a lot.
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  4. #4
    MHF Contributor matheagle's Avatar
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    To be precise F_{7,9} and look up those numbers.
    Remember the mean of an F is near 1, so you know which tail to use in this case.
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