Linear combination of chi-square variables

Discussion in 'Probability' started by Paul, Aug 31, 2011.

  1. Paul

    Paul Guest

    I'd like to know the distribution of a linear combination of two chi-
    square variables. In scanning the literature I've found some general
    results that are relevant but complicated. I'm thinking there might be
    a simpler answer because of some constraints on my specific problem.
    My combination is this:

    V = Uk + f U1

    where Uk is a chi-square distribution with k degrees of freedom,
    U1 is a chi-square distribution with 1 degree of freedom,
    and f<1 is a constant

    Any thoughts on the distribution of V?

    Paul, Aug 31, 2011
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  2. I assume your two chi-squared r.v.'s are independent. You are adding
    independent gammas with different shape parameters. I don't think there
    is much you can say about that. By calculating the convolution
    directly, you may be able to come up with some formula for the density.
    The MGF is t |-> 1/sqrt((1-2t)^k (1-2ft)).
    Stephen J. Herschkorn, Sep 1, 2011
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