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Math Help - determine distribution function and nth moment

  1. #1
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    determine distribution function and nth moment

    Hi, I'm so struggling with this quetsion, can anyone please kindly help me? Thanks a lot.

    Let \alpha, \beta >0 and
    f(x)=\left\{\begin{array}{cc}\frac{\alpha}{\beta}(  \frac{x}{\alpha})^{\alpha -1} \exp(-(\frac{x}{\beta})^\alpha),&\mbox{ if } x>0\\0,&\mbox{otherwise}\end{array}\right
    and suppose the random variable X has density f.

    (a)Determine the distribution function F(x) of X
    (b)Calculate the nth moment E(X^n) for all n=1,... Express it in terms of the Gamma function.


    I did try to integrate PDF to get distribution function, but I end up a constant \frac{\alpha}{\beta}, it must be wrong. Also, I have no idea about the second one. Can anyone please help me? Thanks a lot.
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  2. #2
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    Quote Originally Posted by tsang View Post
    Hi, I'm so struggling with this quetsion, can anyone please kindly help me? Thanks a lot.

    Let \alpha, \beta >0 and
    f(x)=\left\{\begin{array}{cc}\frac{\alpha}{\beta}(  \frac{x}{\alpha})^{\alpha -1} \exp(-(\frac{x}{\beta})^\alpha),&\mbox{ if } x>0\\0,&\mbox{otherwise}\end{array}\right
    and suppose the random variable X has density f.

    (a)Determine the distribution function F(x) of X
    (b)Calculate the nth moment E(X^n) for all n=1,... Express it in terms of the Gamma function.


    I did try to integrate PDF to get distribution function, but I end up a constant \frac{\alpha}{\beta}, it must be wrong. Also, I have no idea about the second one. Can anyone please help me? Thanks a lot.
    (a) Please show all your work so that it can be reviewed.

    (b) Apply the definition and do the integration. Again, please show your work.
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  3. #3
    MHF Contributor matheagle's Avatar
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    Looks like you need a beta where that alpha is in that ratio
    Weibull distribution - Wikipedia, the free encyclopedia
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