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Moment generating function and expectation

  1. Jul 31, 2009 #1
    1. The problem statement, all variables and given/known data
    Let X denote a random variable with the following probability mass function:
    P(j)= 2^(-j), j=1,2,3,.....
    (a) Compute the moment generating function of X.
    (b) Use your answer to part (a) to compute the expectation of X.

    2. Relevant equations
    m.g.f of X is M (t) = E[e^tX]


    3. The attempt at a solution

    I computed the MGF (X) to be: e^t/(2-e^t). I need a suggestion about the next step. I'm confused about the relationship between the MGF and expectation of X.

    Thanks!
     
  2. jcsd
  3. Aug 1, 2009 #2

    tiny-tim

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    Homework Helper

  4. Aug 1, 2009 #3
    That link was very helpful. I completed the problem but I am a little uncertain about my strategy of finding the derivative by parts and the answer:

    X has p.d.f : p(j) = 2^(-j), j=1,2,3,....

    I computed the MGF for X = e^t/(2-e^t)

    Then E[X] = d/dt (e^t/(2-e^t)) [evaluated for t= 0] =

    numerator = (2-e^t)d/dt(e^t) - (e^t)d/dt(2-e^t)
    denominator = (2-e^t)^2

    Evaluating for t=0, the final answer is:
    E[X] = 2

    Is this the correct strategy and answer?

    Thanks!
     
  5. Aug 1, 2009 #4

    tiny-tim

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    Hi BookMark440! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    Looks good! :biggrin:
     
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