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Moment generating functions help

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  1. Apr 9, 2017 #1
    1. The problem statement, all variables and given/known data

    Let X be a random variable with support on the positive integers (1, 2, 3, . . .) and PMF f(x) = C2 ^(-x) .

    (a) For what value(s) of C is f a valid PMF?
    (b) Show that the moment generating function of X is m(t) = Ce^t/(2− e^t) , and determine the interval for t for which it is valid. (You may use your value for C calculated in question 1, if you would like).
    (c) Using the MGF, calculate the expected value and the variance of X.


    3. The attempt at a solution

    a)

    sum from -∞ to ∞ of C/(2^x)=1

    C(1/2+1/4.....)=1
    C=1 as it converges

    b)

    m(t)=E[e^tx]=integral from -∞ to ∞ of ((e^tx)*(2^(-x)))
    =integral from -∞ to ∞ of (e^tx)/(2^x)

    is this the right way to go about calculating it?
     
  2. jcsd
  3. Apr 10, 2017 #2

    Ray Vickson

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    No. For a discrete random variable we have
    $$E f(X) = \sum_x p(x) f(x), $$
    so involves summation, not integration.
     
  4. Apr 10, 2017 #3
    when calculating the sum I get:

    the sum of x=0 to ∞ of (e^tx)/(2^x)=e^t/2

    which is wrong am I still missing something?
     
  5. Apr 10, 2017 #4

    Ray Vickson

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    Yes: you are basically saying that ##\sum_{k=0}^{\infty} r^k = r, ## which is wrong.
     
  6. Apr 10, 2017 #5

    Do I need to see if the series converges or find the partial sum?

    not sure how to start solving it
     
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