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I would like to know if my understanding of the following is correct. This has been tripping me up for a long time now.

Compute $\lim_{x\rightarrow \infty}x^{1-\beta}$.

This is part of a homework problem regarding the expectation of the Pareto distribution. It says everywhere that the expectation is for $\beta>1$ only, but what about $B=1$? Then doesn't the expectation change significantly because the limit at $\beta = 0$ evaluates to 1?

Could someone please clarify my doubt? Thanks.

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  • $\begingroup$ If this is for the purposes of study, please add the self-study tag $\endgroup$
    – Glen_b
    Commented Oct 3, 2013 at 4:38
  • $\begingroup$ Because a request for clarification of this question was made in an answer and no response was received, I have added to the votes to close as unclear, in the sincere hope it will be improved. $\endgroup$
    – whuber ♦
    Commented Oct 3, 2013 at 13:30
  • $\begingroup$ @whuber - it seems to me that he's asking what that limit has to do with the expectation of the Pareto when $\beta = 1$. I agree it's a little unclear, but... to the OP: if what I've written is correct, it would improve your question considerably if you put an explicit statement to that effect in place of the "... clarify my doubt?" line. $\endgroup$
    – jbowman
    Commented Oct 3, 2013 at 13:54
  • $\begingroup$ @jbowman The problems with notation pointed out by Glen_b need fixing, too. $\endgroup$
    – whuber ♦
    Commented Oct 3, 2013 at 14:08
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    $\begingroup$ @whuber No disagreement there. Plus the tag... OK I withdraw my comment, but I'll leave it up as a potentially useful guide to the OP on what to fix. $\endgroup$
    – jbowman
    Commented Oct 3, 2013 at 14:15

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Your value of $\beta$ is a fixed quantity, taken as being known to be $>1$.

I am not sure how $B$ comes in. What is $B$ here? Did you mean $\beta$?

In the case where $\beta=1$ is doesn't have a mean. When $\beta>1$ it does. Find it for $\beta>1$.

There's no issue with 'changing significantly' because $\beta$ doesn't vary in this calculation. (The particular limit in your question exists for $\beta>1$ but $\beta$ is fixed when evaluating tha limit ($x$ is the variable), and is taken to be $>1$ for the purpose of this part.

http://en.wikipedia.org/wiki/Pareto_distribution

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  • $\begingroup$ To extend the answer slightly - the point about the limit is that it shows that the expectation is infinite for $\beta = 1$, as $\lim_{a \to \infty} \int_0^a x^{1-\beta}\text{d}x = \infty$. (Note since $x^- = 0$, technically $x$ has an infinite mean rather than not having a mean at all.) $\endgroup$
    – jbowman
    Commented Oct 3, 2013 at 13:32