Re: infinite moments
From: Glen (glenbarnett_at_geocities.com)
Date: 07/15/04
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Date: 14 Jul 2004 17:25:50 -0700
N.woodman@ucl.ac.uk (Nick) wrote in message news:<54skt7rc7de9@legacy>...
> Is there a generalised property of physical systems that produce f(x)
> with an infinite mean (1st moment)?
You mention f(x) so I assume you're interested in the continuous case
(general case is just as easy, but lets stick to what you started with)
The problem is that \lim(b->oo) \int_0^b x f(x) dx doesn't converge.
> For example, whilst there exists a zeroth moment of f(x)=
> (2/pi)/(1+x^2) = 1, the 1st moment is infinite.
> Might it have been
> possible to recognise this function as one with an infinite mean
> without actually trying to evaluate the mean?
Let c=2/pi.
Well, being a little handwavy, note that "(1+x^2)^-1" as x becomes
large positive behaves like 1/x^2. So x.f(x) is like c/x. Which
you would probably already know doesn't integrate.
Let's split the integral for the expectation into two pieces:
[0,k] and (k,oo) (the first part just to avoid considerations
of our approximation near 0). Take k nice and big.
The actual integral of x.f(x) on the first part is nice and bounded.
Now consider the integral from k to infinity of x.f(x) dx.
The integrand equals c/x - (x.f(x) - c/x).
That second term would converge okay, but the first doesn't.
> I'm asking this rather generally, but the problem I'm thinking about
> relates to specific systems which with a few small changes in
> assumptions either produce finite or infinite first moments; and it
> would be good to relate this behaviour back to the assumptions in a
> rational way.
You need to think about how your x.f(x)'s behave in the tail.
Very loosely, if x.f doesn't go to zero faster than 1/x, the
expectation is going to be infinite - but if you don't put some
restrictions on it to consider sufficiently 'nice' problems, there
are some weird possibilities to worry about. You can construct
infinitely differentiable continuous densities where x.f /doesn't/
converge to zero in the tail faster than 1/x but which still
have finite moments - if you make them 'wiggle' in the right ways.
Glen
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