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Math Help - Dividing probability density functions?

  1. #1
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    Dividing probability density functions?

    I have a problem and I am at a loss:


    Suppose that Z is a standard normal random variable and Y1 and Y2 are χ2 distributed random variable with υ1 and υ2 degrees of freedom, respectively. Further, assume that Z and Y1 and Y2 are independent.
    a) Define W = (Z/SQRT(Y1)). Find E(W) and V(W). What assumptions do you need about the value of υ1?
    b) Define U= Y1/Y2 . Find E(U) and V(U). What assumptions do you need about the value of υ1 and υ2?


    I know the moment generating functions of Z, Y1 & Y2, but I don't know how I am supposed to find the distribution of W considering we are dividing densities... Any help would be vastly appreciated!
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  2. #2
    Moo
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  3. #3
    MHF Contributor matheagle's Avatar
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    These are just multiples of T and F random variables.
    Just multiply and divide by the dfs and you will get your answer in seconds.
    Clearly the mean of the first is zero.
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  4. #4
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    Quote Originally Posted by matheagle View Post
    These are just multiples of T and F random variables.
    Just multiply and divide by the dfs and you will get your answer in seconds.
    Clearly the mean of the first is zero.
    Could you please elaborate? I am not sure I know what you mean by T & F random variables... or dfs. :S
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  5. #5
    MHF Contributor matheagle's Avatar
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    df=degrees of freedom
    I figured you were given this problem because you're studying the F and T distributions.
    There are three ways to attack this.
    1 Derive the densities, which is messy and unnecessary given that you're only asked to obtain the moments.
    2 Find the moments via the Y's, which isn't that hard
    3 Use the F and T moments and obtain the moments of these two in a minute.

    Using the last one...

    U= Y_1/Y_2= {u_1\over u_2}\left({ Y_1/u_1\over Y_2/u_2}\right)= {u_1\over u_2}F_{u_1,u_2}

    So E(U)={u_1\over u_2}E(F_{u_1,u_2})

    and V(U)=\left({u_1\over u_2}\right)^2V(F_{u_1,u_2})

    Now look up the mean and variance of an F, which isn't that hard to derive...
    (thats my second suggested technique)
    http://en.wikipedia.org/wiki/F-distribution

    E(U)= E(Y_1)E(Y_2^{-1})

    where E(Y_1)=u_1 and E(Y_2^{-1}) isn't that hard to obtain either.
    Last edited by matheagle; January 29th 2010 at 04:43 PM.
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  6. #6
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    How do you obtain E(Y^-1)?
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