Does the mean of the Cauchy distribution actually exist?

[Thejas15101998]

I did not understand of the non-existence of variance.
What does it mean?

 

[Dale]

I have never heard of that. Do you have a reference?

 

[Thejas15101998]

I have never heard of that. Do you have a reference?

yes.
Refer to Philip Bevington's book on error analysis , pg 11 last paragraph.

 

[fresh_42]

I assume that either a variance of zero is meant, or that it cannot be defined, e.g. because the expectation value of the variable isn't finite or the variance itself isn't.

 

[micromass]

yes.
Refer to Philip Bevington's book on error analysis , pg 11 last paragraph.

Sorry, but we don't have access to this book. You're going to have to give us all the information you can so we can solve your question.

 

[mathman]

Sometimes a random variable with infinite second moment is said to have no variance.

 

[Stephen Tashi]

The Cauchy distribution has no mean and hence (since the definition of the variance of a probability distribution requires that the mean exists) it has no variance.

 

[micromass]

Can we please stop guessing what the OP means until he gives more information...

 

[Stephen Tashi]

Can we please stop guessing what the OP means until he gives more information...

There would be a lot of stalled threads if we followed that policy consistently.

 

[micromass]

There would be a lot of stalled threads if we followed that policy consistently.

Threads which SHOULD be stalled.

 

[Thejas15101998]

 

[Thejas15101998]

So the above is the reference from the book

 

[fresh_42]

It ("no variance") means "cannot be defined" or "is infinite". There can be several reasons for it as mentioned in previous posts. The author talks about distributions that don't behave mathematically nice. In this context, every distribution is possible and therefore also those, for which the variance cannot be defined (or is infinite). Since it is a very general statement, it cannot be said, what exactly is meant. And it isn't important here anyway.

 

[Thejas15101998]

The Cauchy distribution has no mean and hence (since the definition of the variance of a probability distribution requires that the mean exists) it has no variance.

well yes it is the consequence of its slowly decreasing behavior for large deviations.

 

[mathman]

View attachment 111249

For an experimental distribution, mean and variance can always be computed. I think you need to clarify what you mean when using the terms: average deviation, standard deviation, variance.

 

[StoneTemplePython]

I am new to these forums, but I'll guess that you're looking for a simple non-trivial example that could fit in with the blurb from your text.

Cauchy is bundling a few different things, so I don’t think it’s what you’re looking for (i.e. infinite second moment and no well defined first moment.)

Note that if something is infinite it is said to be in the extended real number system but not part of the real number system. So with respect to (non-extended) real numbers, infinite variance is sometimes referred to as variance not existing. (I think mathman was getting at this.)

For a simple example, take a look at Pareto distribution, with
$$1 < \alpha \leq 2$$.
https://en.wikipedia.org/wiki/Pareto_distribution

In that interval you do have a finite mean, and infinite variance.

Also note that average deviation can be interpreted as a length 1 norm for distance from the mean, whereas variance (or really standard deviation) can be interpreted as a length 2 norm for distance from the mean.

Note that if you are collecting a finite amount of data, your estimates of mean and variance can always be computed and will be finite. But if you are using something like a Pareto distribution with $$\alpha = 1.5$$

what will happen is your mean estimate will approximate the true mean, while your variance estimate will be all over the place. Playing around with simulations may help with intuition here.

 

[Thejas15101998]

 

[Thejas15101998]

This is a snapshot from the page that follows, from the same book.

 

[StoneTemplePython]

If I'm looking to learn something new, I always ask what the simplest non-trivial example is. I strongly encourage you to take a look at the Pareto distribution I linked to above. If you understand how this Pareto works, *then* I would look at your most recent blurb.

The reason is that your picture is Cauchy again. If you scroll down to "Explanation of undefined moments" in its wikipedia entry, here:

https://en.wikipedia.org/wiki/Cauchy_distribution#Explanation_of_undefined_moments

you can get a flavor what is going on. The issue is mathematically speaking, your text is wrong. It confuses the principal value with the Cauchy expected value, which isn't well defined. (Why? As shown in wikipedia, start with the two different finite integral (or even approximate riemann sums) bounds of [-a,a] and [-2a, a] and take a limit as a -> infinity. The result has the same bounds yet different values. This is a subtle mathematical point tied in with one to one mapping, and something that your text does not seem to understand. Put differently, your author is positing [-a,a] without realizing it.)

One solution to this, would come from Jaynes. He would recommend you basically always start with a finite case, state exactly what you mean, and take limits -- people make egregious errors when they start in infinity.

What your book is trying to say is that the second moment $$E[X^2] = \infty$$ $$variance = E[X^2] - E[X]^2 = \infty - ?$$ which is not in the real number system. I'd get a different book, though.

 

[micromass]

Oh wow. The book actually says the mean of the Cauchy distribution exists. That's just sad. I also suggest to throw away the book and get a better one (Taylor).