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Math Help - Moment Generating for Normal

  1. #1
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    Moment Generating for Normal

    Suppose that X~N(Ux,VARx) and that Y~N(Uy,VARy).

    1). Write down the moment generating function Mx(t) = E(e^tx) and My(t) = E(e^ty) of X and Y respectively.

    2). If X and Y are independent random variables, derive the moment generating function of the new random variable W=1.2X + 1.5Y.

    3). Identify the distribution of W, its expected value and its variance.
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  2. #2
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    Hello,
    Quote Originally Posted by Jontan View Post
    Suppose that X~N(Ux,VARx) and that Y~N(Uy,VARy).

    1). Write down the moment generating function Mx(t) = E(e^tx) and My(t) = E(e^ty) of X and Y respectively.
    The mgf of a normal distribution is M_X(t)=\exp\left(U_X t+\sigma^2_X \cdot\frac{t^2}{2}\right)
    M_Y(t)=\exp\left(U_Y t+\sigma^2_Y \cdot\frac{t^2}{2}\right)

    2). If X and Y are independent random variables, derive the moment generating function of the new random variable W=1.2X + 1.5Y.
    M_W(t)=\mathbb{E}\left(e^{tW}\right)=\mathbb{E}\le  ft(e^{1.2tX+1.5tY}\right)=\mathbb{E}\left(e^{1.2tX  }\cdot e^{1.5tY}\right)

    Since X and Y are independent, the expectation of this product is the product of the expectations :

    M_W(t)=\mathbb{E}\left(e^{1.2tX}\right)\cdot\mathb  b{E}\left(e^{1.5tY}\right)=M_X(1.2t)\cdot M_Y(1.5t)=\exp(\dots)

    3). Identify the distribution of W, its expected value and its variance.
    Differentiate the mgf, as you must've been taught
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  3. #3
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    For part 2, would it become: Mw(t) = (e^(1.2Ux + 1.5Uy)t + (1.44VARx + 2.25VARy)(t^2)/2 ?

    For part 3: W~N(1.2Ux + 1.5Uy, 1.44VARx + 2.25VARy)

    Where E(X) = 1.2Ux + 1.5Uy and Var(X) = 1.44VARx + 2.25VARy ?


    Thanks alot!! xxx
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  4. #4
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    Quote Originally Posted by Jontan View Post
    For part 2, would it become: Mw(t) = (e^(1.2Ux + 1.5Uy)t + (1.44VARx + 2.25VARy)(t^2)/2 ?
    Yes

    Except that the parenthesis are not correct.

    It should be \exp\left((1.2U_X+1.5U_Y)t+(1.44\sigma^2_X+2.25\si  gma^2_Y)\cdot\frac{t^2}{2}\right)

    For part 3: W~N(1.2Ux + 1.5Uy, 1.44VARx + 2.25VARy)


    Where E(W) = 1.2Ux + 1.5Uy and Var(W) = 1.44VARx + 2.25VARy ?
    And note that you could've got the expectation and the variance of W directly, by using these properties :
    E(aX)=aE(X)
    E(X+Y)=E(X)+E(Y)
    Var(aX)=aČVar(X)
    Var(X+Y)=Var(X)+Var(Y) if X and Y are independent.

    Thanks alot!! xxx
    You're welcome
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