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Probability density with finite moment?

  1. Mar 11, 2009 #1
    1. The problem statement, all variables and given/known data
    From Hoel, Port, & Stone, Chapter 4, Exercise 9: Construct an example of a density that has a finite moment of order r but has no higher finite moment. Hint: Consider the series [tex] \sum_{k=1}^{\infty} k^{-(r+2)} [/tex] and make this into a density.

    Btw, this is for my own self-study. I don't even know if schools use this book anymore (my edition is from 1971).

    2. Relevant equations
    The r-th moment for a discrete random variable is defined as [tex] \sum_{k}k^r f(k) [/tex] where f(k) is a probability density function [tex] (i.e., \sum_{k}f(k) = 1 ) [/tex]

    3. The attempt at a solution
    If we were to take the Hint and pretend f is a probability density defined by [tex] f(k) = k^{-(r+2)} [/tex], then I see that a moment of order > r would result in something greater than a harmonic series, which is divergent. But I don't see how to turn f into a probability density function. I know [tex] \sum_{k=1}^{\infty} k^{-2} = \pi / 6[/tex], but I'm pretty sure there are no "convenient" solutions for [tex] \sum_{k=1}^{\infty} k^{-(r+2)} [/tex] where r is an arbitrary positive integer. If there were, than I could just normalize the series to 1 by dividing it, e.g., [tex] \frac{6}{\pi} \sum_{k=1}^{\infty} k^{-2}[/tex] qualifies as a probability density for r=0.

    Any help would be much appreciated! A hint or two would be ideal. All of the exercises in this book have been relatively straightforward up to this point, so I feel like this question should not be that difficult. Perhaps I am just misunderstanding the question.
     
  2. jcsd
  3. Mar 12, 2009 #2
    Ah, figured it out. I was looking for a nice solution when the raw form would suffice... just divide the series above by itself. So the probability density function would be:
    [tex]
    \frac{1}{\sum_{k=1}^{\infty} k^{-(r+2)}} \sum_{k=1}^{\infty} k^{-(r+2)} [/tex]
    [tex]
    = \sum_{k=1}^{\infty} \frac{k^{-(r+2)}}{\sum_{k=1}^{\infty} k^{-(r+2)}} [/tex]

    This converges to 1 for r >= 0. And moments of order > r would diverge.
     
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