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Thread: MGF, Moment-Generating Function

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    MGF, Moment-Generating Function




    Let X1 and X2 be two independent random variables.
    Let X1 and Y = X1 + X2 be χ2(r1) (Chi Square) and χ2(r),
    respectively, where r1 < r.
    (a) Find the mgf of X2.
    (b) What is its distribution?

  2. #2
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    Re: MGF, Moment-Generating Function

    Do you know the mgf of a chi square random variable?
    Do you know the mgf of the sum of two random variables?

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    Re: MGF, Moment-Generating Function

    Chi-squared or χ2
    : Let Z1, Z2, , Zn be independent standard normal
    RVs. Let X = Sum Zi^2 i=1...n

    Then X has the chi-squared distribution with n degrees of freedom. It can
    be shown that this is the gamma distribution with α = n/2 and β = 1/2. So
    the pdf is
    f(x|n) = 1/(2^(n/2)(Γ(n/2)) x^(n/2−1)e^(−x/2), x ≥ 0
    E[X] = n, V ar(X) = 2n, M(t) = (1/(1-2t))^(n/2)
    Note that the sum of independent chi-squared is again with chi-squared with
    the number of degree of freedom adding.

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    Re: MGF, Moment-Generating Function

    Quote Originally Posted by BGM View Post
    Do you know the mgf of a chi square random variable?
    Do you know the mgf of the sum of two random variables?
    I am not sure about the first question. But I know about the mgf of the sum of two random variable. It is equal of multiplications of the mgf of each random variable,

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    Re: MGF, Moment-Generating Function


    You have answered the first question in #3. As you also understand the second question, now you may can to form a equation relating the mgfs of Y, X_1, X_2

  6. The Following User Says Thank You to BGM For This Useful Post:

    aytajalli (02-01-2015)

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