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Math Help - Moment generating functions

  1. #1
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    Moment generating functions

    Independent random variables
    X; Y and Z are identically distributed. Let

    W
    = X+Y. The moment generating function of W is MW(t) = (0.7+0.3*e^t)^6:

    Find the moment generating function of V = X + Y + Z:

    I have started it with
    Mv(t)=E[e^(X+Y+Z)*t]
    =E[(e^(X+Y)*t)*(e^Z)*t]
    since X,Y,Z are independent .we can write
    Mv(t)=E[e^((X+Y)*t)]*E[e^(Z*t)]
    =Mw(t)*Mz(t)
    Here Mw(t) is given ,but how do I find out Mz(t)..
    help me to solve this
    Thanks in advance...
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  2. #2
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    Quote Originally Posted by Pushpa View Post
    Independent random variables

    X; Y and Z are identically distributed. Let


    W
    = X+Y. The moment generating function of W is MW(t) = (0.7+0.3*e^t)^6:
    Find the moment generating function of V = X + Y + Z:

    I have started it with
    Mv(t)=E[e^(X+Y+Z)*t]
    =E[(e^(X+Y)*t)*(e^Z)*t]
    since X,Y,Z are independent .we can write
    Mv(t)=E[e^((X+Y)*t)]*E[e^(Z*t)]
    =Mw(t)*Mz(t)
    Here Mw(t) is given ,but how do I find out Mz(t)..
    help me to solve this
    Thanks in advance...
    There is a theorem that says that if X_1, X_2, .... X_n are independent random variables with mgf's m_{X_1}(t), m_{X_2}(t), .... m_{X_n}(t) then the mgf of U = X_1 + X_2 + .... + X_n is m_U(t) = m_{X_1}(t) \cdot m_{X_2}(t) \cdot .... \cdot m_{X_n}(t).

    Using this theorem it seems to me that the since X, Y and Z are i.i.d. random variables then the mgf of V = X + Y + Z is  m_V = (0.7+0.3 e^t)^9 (and that, in fact, X, Y and Z are binomial random variables with n = 3 and p = 0.3).
    Last edited by mr fantastic; July 1st 2010 at 09:44 PM. Reason: fixed LaTeX
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  3. #3
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    I got it....
    Mz(t)=(0.3+0.7*e^t)^3 (since X,Y,Z are identically distributed.)
    and therefore Mv(t) = ((0.3+0.7*e^t)^9)
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