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Stability of the gaussian under addition and scalar multiplication

  1. Jun 15, 2011 #1
    If i have the mgf of X and the mgf of Y where X~N(mx,vx) and Y~N(my,vy) and X and Y are independent ,
    then if i want to show that aX +bY ~ N(amx+bmy , a^2vx+b^2vy) how would i do this - need to be able to do the convolutions way and the mgf's way,
    for the mgfs way is it just, mgf(ax+by) = mgf(ax) . mgf(bx) if so how do you find mgf(ax) ?

    Thanks
     
  2. jcsd
  3. Jun 15, 2011 #2

    micromass

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    Hi stukbv! :smile:

    Sadly, there is no way to easily calculate the mgf of aX when just knowing X (or at least, I don't know such a way). The easiest is by showing in general that the mgf of [itex]X\sim N(\mu,\sigma^2)[/itex] is

    [tex]M(t)=e^{t\mu+\frac{1}{2}\sigma^2t^2}[/tex]

    Then the distribution of aX is

    [tex]M(at)=e^{at\mu+\frac{1}{2}\sigma^2a^2t^2}[/tex]

    And what kind of distribution is this?
     
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