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Math Help - moment-generating function

  1. #1
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    moment-generating function

    We went over this material once in class and we have a bunch of problems that we have to do, and I still don't know if I understand the concept fully. One of the problems is:

    Find the moment-generating function of the discrete random variable X that has the probability distribution
    f(x) = 2(1/3)^x for x = 1, 2, 3, ...
    and use it to determine the values of mu1' and mu2'.

    So I tried to use the definition, which says that Mx(t) = E(e^(tx)) = Integral from -INF to INF of (e^(tx))*f(x)dx.

    I got the Integral from 1 to INF of (e^(tx))*(2(1/3)^x)dx, but now I am lost... any help would be GREATLY appreciated.
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  2. #2
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    The moment generating function definition you have used is for a continuous random variable with probability distribution f(x).

    In every case we can use the definition: m_X(t) = E[e^{tX}] .

    So in your case:
    m_X(t) = 2 \left(\frac{1}{3}e^t + \left(\frac{1}{3}\right)^2 e^{2t} + ...\right) = \frac{2}{3}e^{t} \left(1+\frac{1}{3}e^t + \left(\frac{1}{3}e^t \right)^2 + \left(\frac{1}{3}e^t\right)^3+... \right)
    =\frac{\frac{2}{3}e^t}{1-\frac{1}{3}e^t} = \frac{2e^t}{3-e^t}
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  3. #3
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    I see your first two steps, but how did your third step become so simplified? Could you show me the step in between that? Thanks!
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  4. #4
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    This is the geometric series:

    \sum_{i=0}^\infty ar^{i} = \frac{a}{1-r}
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  5. #5
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    Thank you so much! That really helps a lot. Now I will see if I can do the rest on my own.
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  6. #6
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    Also, could I ask, anyone, what program you have on your computer that lets you type mathematically?
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  7. #7
    Senior Member topher0805's Avatar
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    Quote Originally Posted by TheHolly View Post
    Also, could I ask, anyone, what program you have on your computer that lets you type mathematically?
    It's called Latex and it isn't a program. It's built right into this site so any user can use it.

    You see the button beside the YouTube button when you make a post? That button wraps the selected text with [tex] tags.

    You can also click on the math equations you see on here to find out how they were written. For example, to write:

    <br />
\sin {\sqrt {4x^2}} = \log {\frac {1}{2}}

    I would have to write:

    \sin {\sqrt {4x^2}} = \log {\frac {1}{2}}

    And then wrap it with math tags.
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  8. #8
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    Sorry, guys, but I still can't figure out mu1' or mu2'. Can anyone help???
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  9. #9
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    Quote Originally Posted by TheHolly View Post
    Sorry, guys, but I still can't figure out mu1' or mu2'. Can anyone help???
    By mu1' do you mean E(X)?
    By mu2' do you mean E(X^2)?

    If so, note that


    E(X^n) = \left[ \frac{d^n M_X}{dt^n} \right]_{t = 0}.


    that is, evaluate the nth derivative of M_X at t = 0.

    Solve for n = 1 (I get 3/2) and n = 2 (I get 3).
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