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Math Help - Moment Generating Functions

  1. #1
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    Moment Generating Functions

    Ok... this one is taking me forever so I could use some help...

    Let X be a random variable with pdf f(x) and mgf M(t). Suppose f is symmetric about 0, (f(-x) = f(x)). Show that M(-t) = M(t).

    If someone could figure this out for me I would really appreciate it.. I would really just love to know where to start!
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  2. #2
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    Quote Originally Posted by jandrewross View Post
    Ok... this one is taking me forever so I could use some help...

    Let X be a random variable with pdf f(x) and mgf M(t). Suppose f is symmetric about 0, (f(-x) = f(x)). Show that M(-t) = M(t).

    If someone could figure this out for me I would really appreciate it.. I would really just love to know where to start!
    Start with the definition of the moment generating function:

     M_X(t)=E(e^{tX}) =\int_{-\infty}^{\infty} e^{tx} p(x) dx

    then:

     M_X(-t)=E(e^{-tX}) =\int_{-\infty}^{\infty} e^{-tx} p(x) dx

    Now let y=-x, then:

     M_X(-t)=E(e^{-tX}) =\int_{\infty}^{-\infty} e^{ty} p(-y) (-1)dy <br />
=\int_{-\infty}^{\infty} e^{ty} p(y) dy=M_X(t)<br />

    RonL
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