I believe I read today a phrase which went something like this:

If a distribution has a mean and a variance ...

So I guess that means some distributions do not have means or variances?

I fiend that a bit difficult to understand - for example:

I could get data which is said to follow such a mean-less and variance-less distribution and as an estimation compute the mean and variance of such a data set.

could someone help shed some light on what I am missing?

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    $\begingroup$ The Cauchy distribution is a famous distribution that has neither a mean nor a variance (no finite moments in general). This post explains very nicely why not. You can, of course compute the sample mean and variance. $\endgroup$ – COOLSerdash Sep 15 '13 at 21:37
  • $\begingroup$ I just read the post - fascinating, fascinating $\endgroup$ – user1172468 Sep 15 '13 at 21:40
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    $\begingroup$ The mean and/or variance don't exist when the limits implied by the improper integrals don't exist; e.g. for the mean, $\lim_{a,b\rightarrow\infty}\int_{-a}^b x\, dF$ has to exist for the mean to exist (for a continuous density $dF = f(x)dx$). For a non-Cauchy example, consider the discrete distribution $P(X=x)= \frac{6}{\pi^2x^2};\,x=1,2,3,...$, a particular case of the zeta distribution; in this case neither variance nor mean exist (it's the case that if the mean doesn't exist, the variance won't either, but the mean can exist when the variance doesn't). $\endgroup$ – Glen_b -Reinstate Monica Sep 16 '13 at 0:53
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    $\begingroup$ The wiki article en.wikipedia.org/wiki/… explains well what are you estimating by calculating the sample mean, and how "good" is this estimation. $\endgroup$ – Alecos Papadopoulos Sep 16 '13 at 23:55
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    $\begingroup$ See this related question. The sample mean and variance will always exist (the sample consists of a finite collection of numbers, for which the sample quantities will be well-defined), it's the things that they estimate that are not finite (either infinite or undefined), and as such the sample estimates can't really convey useful information about the population. $\endgroup$ – Glen_b -Reinstate Monica Feb 16 '17 at 22:00

The canonical example of a distribution with no mean (and hence no variance) is the Cauchy distribution.

True, you could compute the sample mean for a sample of data from a Cauchy distribution, but you would have to interpret such a sample mean carefully; increasing the sample size would not correspond to getting a better estimate of the TRUE mean, because there is no true (i.e., large sample) mean.


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