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Joint pdf question and mgf

  1. Apr 1, 2012 #1
    Hi guys,

    I'm really stuck on the following questions, not sure as to how to approach it:

    Let X and Y be random variables for which the joint pdf is as follows:

    f(x,y) = 2(x+y) for 0 <= x <= y <= 1
    and 0 otherwise.

    Find the pdf of Z = X + Y

    And also:

    Suppose that X is a random variable for which the mgf is as follows:

    /u(t) = e^(t^2 + 3t) for minus infinity < t < infinity

    Find the mean and variance for X.
    I know that the answers are 3 and 2 respectively, but was unsure how they got to the answer, do I need to integrate by parts?

    Any help would be appreciated! Thanks guys :)
  2. jcsd
  3. Apr 2, 2012 #2


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    Science Advisor

    I'll address the second question only. The moments are obtained from the moment generating function by simply taking derivatives and setting t = 0. As you must be aware, the variance is the second moment minus square of first moment.
  4. Apr 3, 2012 #3
    Figured out second question now, pretty straightforward in hindsight. Any help on the first one? ;)
  5. Apr 3, 2012 #4


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    Science Advisor

    Have you tried a transformation? Let u = x + y. Now use that transformation to get a integral in terms of u, take into account limits and then use transformation theorem to relate g(u) = 2(x+y) = 2u to another PDF f(u) which represents the distribution of Z.
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