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Math Help - The poisson....

  1. #1
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    The poisson....

    The problem as stated is Let X have a poisson distribution w/ lambda = n.

    If Y= (X-n)/Sqrt(n), find the MGF for Y.

    I know that M(t) for X is given by e^n((e^t)-1). I cannot figure out how to make the jump to find the MGF for Y. I have tried finding the expected value of e^ty but what I get makes no sense. Help.....
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  2. #2
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    Quote Originally Posted by roresc View Post
    The problem as stated is Let X have a poisson distribution w/ lambda = n.

    If Y= (X-n)/Sqrt(n), find the MGF for Y.

    I know that M(t) for X is given by e^n((e^t)-1). I cannot figure out how to make the jump to find the MGF for Y. I have tried finding the expected value of e^ty but what I get makes no sense. Help.....
    Y = \frac{1}{\sqrt{n}} X - \sqrt{n}.

    You're expected to know that if Y = aX + b then m_Y(t) = e^{bt} M_X(at).

    This theorem follows easily from the linearity of expectations.
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  3. #3
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    Hello,
    Quote Originally Posted by roresc View Post
    The problem as stated is Let X have a poisson distribution w/ lambda = n.

    If Y= (X-n)/Sqrt(n), find the MGF for Y.

    I know that M(t) for X is given by e^n((e^t)-1). I cannot figure out how to make the jump to find the MGF for Y. I have tried finding the expected value of e^ty but what I get makes no sense. Help.....
    Go back to the definition of the mgf :
    M_Y(t)=\mathbb{E}(e^{tY})=\mathbb{E}(e^{\frac{Xt}{  \sqrt{n}}-\frac{1}{\sqrt{n}}})\mathbb{E}(e^{\frac{Xt}{\sqrt{  n}}} e^{-\frac{1}{\sqrt{n}}})

    But e^{-\frac{1}{\sqrt{n}}} is a constant. And since for any constant a, \mathbb{E}(aX)=a \mathbb{E}(X), we have :

    M_Y(t)=e^{-\frac{1}{\sqrt{n}}} \cdot \mathbb{E}(e^{X \cdot \frac{t}{\sqrt{n}}})=e^{-\frac{1}{\sqrt{n}}} \cdot M\left(\tfrac{t}{\sqrt{n}}\right)


    Edit : woops too late...Again ><
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  4. #4
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    Thanks to both.

    Thanks to both for the help. Forgot my rules of exponents here. Things are now clear. Thanks again...
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