Re: Moments


Jani Huhtanen wrote:
Andor wrote:


Jani Huhtanen wrote:
Andor wrote:

Jani Huhtanen wrote:
Andor wrote:

John Herman wrote:
I don't think that skewnew is the name for the third moment or
kurtosis for
the fourth. Skew and the kurtosis are the names for parameters of
distributions relative to the Gaussian.

All those "moments" with a name aren't really moments in anycase,
since they are centered. For example, for the Cauchy distribution
there exists no variance, but a second moment (which is infinite).


So you're saying that a second moment exists although the integral
diverges? In fact any moments mu_k, where k>=1, diverge for Cauchy
distribution and are quite often declared to be undefined.

The difference lies in the ability to specify a limit, even if it is
infinity. For example

limit_{x->inf} x = inf.

However,

limit_{x->inf} x sin(x)

does not exist. Check out the section "Why the second moment of the
Cauchy distribution is infinite" in

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


Interesting. I see the difference. Mathworld, however, defined all the
moments (except 0th) undefined. I begin to wonder, why is the mean so
hard to define? The distribution is symmetric, thus doesn't it follow
automatically that the mean is at the point of symmetry. In this case,
the maxima of the distribution. In fact, parameter x0 seems to define the
place of the maxima.

Yet another pretender to the "mean" could be the 50% percentile
(median) of the distribution. I can easily think of an asymmetric
distribution where the mean, the maximum of the density and the median
are all diffierent.


Of course. But isn't Cauchy symmetric?

Yes - I was just giving examples where definitions such as point of
symmetry, maximum of density, median, etc. do _not_ have the same
answer as the mean (and are thus not universally applicable as
surrogate for the mean).


What am I missing?

Only the mean is defined in such a way that the large number laws in
probability theory are valid.


I must be dumb, but how has law of large numbers anything to do with this?
We are not estimating the mean from a population as far as I can tell.

I thought you were asking why we define the mean the way it is defined.
What was the question again?

.

Relevant Pages

  • Re: Moments
    ... All those "moments" with a name aren't really moments in anycase, ... For example, for the Cauchy distribution ... distribution and are quite often declared to be undefined. ... automatically that the mean is at the point of symmetry. ...
    (comp.dsp)
  • Re: Moments
    ... The distribution is symmetric, thus doesn't it follow ... automatically that the mean is at the point of symmetry. ... the maxima of the distribution. ... int_{0}^{infinity} x fdx = infinity. ...
    (comp.dsp)
  • Re: Moments
    ... All those "moments" with a name aren't really moments in anycase, ... for the Cauchy distribution there ... but a second moment (which is infinite). ...
    (comp.dsp)
  • Re: Moments
    ... Skew and the kurtosis are the names for parameters of ... All those "moments" with a name aren't really moments in anycase, ... For example, for the Cauchy distribution ... distribution and are quite often declared to be undefined. ...
    (comp.dsp)
  • Re: Moments
    ... The distribution is symmetric, thus doesn't it follow ... automatically that the mean is at the point of symmetry. ... maxima of the distribution. ... int_{0}^{infinity} x fdx = infinity. ...
    (comp.dsp)