The Cauchy (as a whole) has undefined expectation. But if you look at any finite range of the distribution, this subset makes a finite, real-valued contribution towards the overall expectation.

This may seem obvious, but it’s related to

@Tal’s: question of how the idea came to me. Most of the time, if you are looking at real-world data, it’s not going to follow any exact, known distribution. At best it will approximate something recognizable. Also, you’re going to be looking at some finite set of data, and presumably the values will be finite. If you look at your data with a histogram or kernel density plot, you’ll see some rough variation on a the standard density plots. This will works even if you have a sample that comes from one of those wickedly stable distributions with no calculable moments. Looking at the chart will give little evidence that you may be in for a nasty surprise if you try to predict future values or a population mean on the basis of your sample. But if you plot your (empirical) density as I am suggesting, in terms of values of x versus that value’s contribution to total value, then you see right away 1. If something strange is going on and 2. What parts of the distribution contribute the most to your average (which are the most influential).

It seems to me that this is really important information to have, and you should get it right away in visual form.

@efrique:

I was referring to the Standard Normal distribution, which has variance equal to standard deviation equal to 1. Sorry if I wasn’t clear about drawing that out. I added “central” to the sentence.

which in the case of the Normal isn’t the mean, but rather the second moment (plus or minus)

The second (central) moment is the variance. The points you’re referring to are one standard deviation either side of the mean.

Where did the idea came to you from ?

p.s: consider installing the plugin “subscribe to comments”

Cheers,
Tal