Does the mean of the Cauchy distribution actually exist?

In summary: I cannot answer this question without knowing more about the context.I cannot answer this question without knowing more about the context.
  • #1
Thejas15101998
43
1
I did not understand of the non-existence of variance.
What does it mean?
 
Physics news on Phys.org
  • #2
I have never heard of that. Do you have a reference?
 
  • #3
Dale said:
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.
 
  • #4
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.
 
  • #5
Thejas15101998 said:
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.
 
  • #6
Sometimes a random variable with infinite second moment is said to have no variance.
 
  • Like
Likes FactChecker
  • #7
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.
 
  • Like
Likes Auto-Didact
  • #8
Can we please stop guessing what the OP means until he gives more information...
 
  • #9
micromass said:
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.
 
  • Like
Likes Auto-Didact
  • #10
Stephen Tashi said:
There would be a lot of stalled threads if we followed that policy consistently.

Threads which SHOULD be stalled.
 
  • #11
upload_2017-1-8_8-16-11.png
 
  • #12
So the above is the reference from the book
 
  • #13
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.
 
  • #14
Stephen Tashi said:
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.
 
  • #15
Thejas15101998 said:
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.
 
  • #16
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.
 
  • Like
Likes Dale
  • #17
upload_2017-1-9_8-2-8.png
 
  • #18
This is a snapshot from the page that follows, from the same book.
 
  • #19
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.
 
Last edited:
  • Like
Likes Thejas15101998
  • #20
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).
 

1. What is "No existence of Variance"?

"No existence of Variance" refers to a situation where there is no difference or variation between two or more groups or samples.

2. How is "No existence of Variance" measured?

"No existence of Variance" is measured using statistical tests such as the F-test or ANOVA, which compare the variances of the groups or samples in question. If the result of the test is not statistically significant, it indicates that there is no difference in variance between the groups.

3. What does "No existence of Variance" mean in terms of data analysis?

In data analysis, "No existence of Variance" means that there is no evidence to suggest that the groups or samples being compared are different from each other. This could be due to chance or random sampling error.

4. What are some possible reasons for "No existence of Variance"?

"No existence of Variance" can occur due to various reasons such as the groups being truly homogeneous, the sample size being too small to detect differences, or the data being collected from a population with low variability.

5. How does "No existence of Variance" impact research findings?

"No existence of Variance" can impact research findings as it indicates that there is no significant difference between the groups being compared. This may lead to the rejection of the research hypothesis and a conclusion that there is no relationship between the variables being studied.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
987
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
280
  • Set Theory, Logic, Probability, Statistics
Replies
28
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
78
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
880
Back
Top