10

I believe $p[x]$ is a probability distribution, where

\begin{equation} p[x] = \frac{1}{\pi (1+x^2)} \end{equation}

since it's positive everywhere and integrates to 1 on $-\infty, \infty$.

The mean is 0 by symmetry, even though integrating $xp[x]$ on $-\infty, \infty$ does not converge. This is "suspicious" since $p[x]$ is supposed to be a probability distribution, but reasonable because $xp[x]$ is $O(1/x)$ which is known to diverge.

The bigger problem is in computing the standard deviation. Since $x^2 p[x]$ also diverges, since $x^2 p[x]$ is $O(1)$.

If this isn't a probability distribution, why not? If it is, is its standard deviation infinite?

The cumulative distribution function is $\arctan[x]/\pi$ if that helps.

Someone mentioned this might be a gamma distribution, but that isn't clear to me.

csgillespie
  • 11,849
  • 9
  • 56
  • 85
  • 1
    @user1566: I formatted your equations using LaTex. Would you double check that I didn't introduce any errors? – csgillespie Nov 09 '10 at 21:31
  • Thanks, the problem is solved, so no longer a biggie, but, yes, everything looks OK. –  Nov 09 '10 at 22:11
  • The mean of a Cauchy is *not* zero. In fact, it doesn't exist. Thus, neither does any of its central moments. – cardinal Apr 24 '11 at 18:55
  • my answer to a related question can be found here. http://stats.stackexchange.com/questions/232967/what-makes-the-mean-of-some-distributions-undefined/232976#232976 – Haitao Du Jan 21 '17 at 01:01

2 Answers2

12

To answer your question title: Yes, a probability distribution can have infinite standard deviation (see below).

Your example is a special case of the Cauchy distribution whose mean or variance does not exist. Set the location parameter to 0 and the scale to 1 for the Cauchy to get to your pdf.

4

The Cauchy distribution doesn't have a mean or variance, in that the integral doesn't converge to anything in $[-\infty,\infty]$. However, a distribution like $f(x)=\frac{2}{x^3}$ on $[1,\infty)$ has a mean, but the standard deviation is infinite.

Alex R.
  • 13,097
  • 2
  • 25
  • 49
  • Of course, in real measurement situations we don't get infinite standard deviations but becomes large compared to the mean value, i.e., the co-efficient of variation is quite large, as it happens in some population growth models or mutation models. It is measurement nightmare, as prediction becomes problematic. – Sukii May 16 '21 at 18:49