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Math Help - Moment generating function and distribution function?

  1. #1
    DCU
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    Moment generating function and distribution function?



    I got part a out easily enough. I'm having big problems on how to do part b and c. I'm like this . Thanks for any help
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  2. #2
    Flow Master
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    Quote Originally Posted by DCU View Post


    I got part a out easily enough. I'm having big problems on how to do part b and c. I'm like this . Thanks for any help
    (b) I suppose they want you to calculate the cdf of X: F(x) = \int_0^x \lambda e^{-\lambda u} \, du.

    This should be a simple integral for you to calculate.


    (c) Apply the definition: M_X (t) = E\left(e^{tX}\right) = \int_0^{+ \infty} e^{tx} \lambda e^{-\lambda x} \, dx = \lambda \int_0^{+ \infty}e^{-(\lambda - t) x} \, dx .

    Again, this should be a simple integral for you to calculate. What happens if \lambda - t < 0 ....?
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  3. #3
    DCU
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    Quote Originally Posted by mr fantastic View Post
    (b) I suppose they want you to calculate the cdf of X: F(x) = \int_0^x \lambda e^{-\lambda u} \, du.

    This should be a simple integral for you to calculate.


    (c) Apply the definition: M_X (t) = E\left(e^{tX}\right) = \int_0^{+ \infty} e^{tx} \lambda e^{-\lambda x} \, dx = \lambda \int_0^{+ \infty}e^{-(\lambda - t) x} \, dx .

    Again, this should be a simple integral for you to calculate. What happens if \lambda - t < 0 ....?
    if \lambda - t < 0, the value of the integral will go to infinity?
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  4. #4
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    Quote Originally Posted by DCU View Post
    if \lambda - t < 0, the value of the integral will go to infinity?
    So that tells you the answer to the last part of (c).
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  5. #5
    DCU
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    Quote Originally Posted by mr fantastic View Post
    So that tells you the answer to the last part of (c).
    so when lambda is greater than t?
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